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Digitized by the Internet Archive 
in 2011 with funding from 
The Library of Congress 



http://www.archive.org/details/measurementsofsoOOwood 



MEASUREMENTS OF SOME 

ACHIEVEMENTS IN 

ARITHMETIC 



BY 

CLIFFORD WOODY, Ph.D. 



TEACHERS COLLEGE, COLUMBIA UNIVERSITY 
CONTRIBUTIONS TO EDUCATION. NO. 80 



PUBLISHED BY 

JSmtli^m (SolUgr, (Snl«mbta Unttt^rsftg 

NEW YORK CITY 

1916 



Monograph 






Copyright, 1916, by Clifford Woody 



JAN 27 1917 

©CI A 4 57370 



CONTENTS 

PART I 
I. Introduction i 

II. The Arithmetic Scales and Their Uses.. 3 

1. Directions for Administering the Tests 

2. Directions for Scoring the Tests 

3. Directions for Determining the Class Score 

4. Tentative Standards of Achievements 

III. The Value and Uses of the Scales 23 

IV. The Limitations of the Scales 24 

PART II 
I. The Derivation of the Scales 25 

1. History of the Scales 

2. Probable Error (P.E.) Taken as the Unit of 

Measure 

3. Scaling the Problems in Addition for Each 

Grade 

4. Measuring the Distance Between the Grades 

5. Location of the Zero Point 

6. Referring All the Problems to Zero 

II. Tables of Crude Data From Which Scales Were 

Developed 55 



ACKNOWLEDGMENTS 

The author wishes to acknowledge his indebtedness to those 
whose aid has made this study possible. He wishes to thank 
the superintendents, principals, and teachers who so willingly 
cooperated in the collection of the data. He feels especially 
indebted to Professor George D. Strayer, Professor Edward 
L. Thorndike, and Dr. Marion Rex Trabue for their valuable 
suggestions and criticisms, which have given to the study what- 
ever merit it possesses. 

C. W. 



MEASUREMENTS OF SOME ACHIEVEMENTS 
IN ARITHMETIC 

PART I 

Section i. INTRODUCTION 

The purpose of this monograph is to set forth the results of 
an attempt to derive a series of scales in the fundamental opera- 
tions of arithmetic. Thus the problem is closely related to the 
general movement for the measurement of educational products 
by means of objective scales. The method followed in the 
development of these scales is most clearly related to the methods 
used by Dr. Buckingham ^ in the development of his Spelling 
Scale and by Dr. Trabue in the Completion-Test Language 
Scales.^ In the development of these scales the fundamental 
idea was to derive a series of scales which would indicate the 
type of problems and the difficulty of the problems that a class 
can solve correctly. Accordingly, each of the scales is composed 
of as great a variety of problems as the fundamental operations 
can well permit. These problems, beginning with the easiest 
that can be found, gradually increase in difficulty until the last 
ones in each series are so difficult that only a relatively small 
percentage of the pupils in the eighth grade are able to solve 
them correctly. In the determination of the relative difficulty 
of these problems, the relative per cents of correct answers 
obtained by submitting them to large numbers of school children 
were taken as a basis. 

Two distinct series of scales in each of the fundamental opera- 
tions have been derived. Series B contains only about half as 
many problems as Series A. Series A thus has a greater power 
of diagnosing the weaknesses of a class and is recommended 
where there is ample time for testing. Series B was derived 

1 Buckingham, B. R., Spelling Ability, Its Measurement and Dis- 
tribution, 1913. 

2 Trabue, Marion Rex, Completion-Test Language Scales, 1916. 

I 



2 Measurements of Some Achievements in Arithmetic 

especially for use where the amount of time that can be devoted 
to measuring is very limited. 

Part I of this monograph is devoted especially to the scales 
and their uses. Specific directions for administering the tests 
and scoring the results are given in detail. A statement of the 
values and limitations of the scales is also given in this part. 

Part II deals with the history and the method of the deriva- 
tion of the scales. It also includes many tables of crude data 
from which the scales were developed. 



Section IL THE ARITHMETIC SCALES AND THEIR 

USES 

I. Directions for Administering the Tests 

These scales are useful as measures of achievement in the 
fundamentals of arithmetic either of a class or of a whole school 
system. Series A is more valuable when the amount of time for 
testing is plentiful. Series B was especially constructed for use 
in measuring school systems where the amount of time for 
testing purposes is limited. Both series of tests are adminis- 
tered in the same way. 

The Addition and Subtraction Scales can be used in grades 
two to eight inclusive ; the Multiplication and Division Scales, 
in grades three to eight inclusive. These scales may be sub- 
mitted in any order to the pupils. They may be given in imme- 
diate succession or with such intervals of time intervening as 
is most convenient. In the development of the scales subtrac- 
tion and multiplication were given in succession on one day and 
addition and division on the next day. The writer recommends 
that for Series B all tests be given in succession. 

If the measurements by these scales are to be valid and com- 
parable, it is necessary that the same standard of procedure be 
followed in giving the tests and in scoring the results as was 
followed in the original development of the scales. The same 
individual should give all of the different tests. He should give 
the same instructions to every class. He should have the same 
manner in each class room. In giving the " specific directions " 
to the class he should use as nearly as possible the same emphasis 
and intonation. He should not stress one part of the directions 
more than another part. 

It is highly important that the teacher or the one in charge of 
the room remain silent (saying nothing to the children indi- 
vidually or collectively during the time of giving the tests). 

When ready to distribute the tests, place one face downward 
on each desk. Insist that the pupils do not turn the papers 

3 



Measurements of Some Achievements in Arithmetic 



Series A^ 
ADDITION SCALE 
Name 



Are yon a bov 


or girl? 


In what, prarlp are von? 




(1) 


— — — J 
(2) 


(3) 


(4) 


(5) (6) 


■ o- 


(7) 


(8) (9 


2 


2 


17 


53 


72 60 


3 + 1= 2+5 + 1= 20 


3 


4 
3 


2 


45 


26 37 






10 

2 

30 




























25 


(10) 


(11) 


(12) 


(13) 


(14) 


(15) 


(16) (17) 


(18) ~~ 


21 


32 


43 


23 


25+42 = 


100 


9 199 


2563 


33 


59 


1 


25 




Zd> 


24 194 


1387 


35 


17 


2 


16 




45 


12 295 


4954 


— 




13 






201 

46 


15 156 
19 


2065 


(19) 


(20) 


(21) 


(22) 


(23) 


(24) 


(25) 


$ .75 


$12.50 


$8.00 


547 i+| = 


4.0125 


1 + 1+1+1 = 


1.25 


16 


.75 


5.75 


197 




1.5907 ' 




.49 


15 


.75 


2.33 

4.16 

94 


685 
678 




4.10 
8.673 




















6.32 


393 
525 
240 






























152 








(26) 


(27) 


(28- 


) (29) 


(30) 


(31) 


(32) 


m 


\+\+h = 


34.1 ■ 


= 4f 


2h 


113.46 


i+h+\ = 


m 








2i 


61 


49.6097 




m 








Si 


3f 


19.9 




371 












9.87 








(34) 




(35) 




.0086 
18.253 
6.04 




(33) 


(36) 


(37) 


.49 




\+i- 


= 


2 ft. 6 in. 




2 yr. 5 mo. 


16^ 


.28 








3 ft. 5 in. 




3 yr. 6 mo. 


121 


.63 








4 ft. 9 in. 




4 yr. 9 mo. 


2U 


.95 












5 yr. 2 mo. 


32f 




1.69 












6 yr. 7 mo. 




99 
















. £t£t 

.33 
















.36 
















1.01 
















.56 
















.88 








(38) 






.75 






25.091 + 100.4+25+98.28 + 19.3614 = 




.56 
















1.10 
















.18 
















.56 


































^The scales are printed in large type, on separate sheets, 8|" x 11", with 
ample space for the insertion of answers. 



The Arithmetic Scales and Their Uses 

Series A 
SUBTRACTION SCALE 

Name 

When is your next birthday? How old will you be?. 

Are you a boy or girl? In what grade are you? 



(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 

8 6 2 9 4 11 13 59 78 7 — 4= 76 

5 1 3 4 7 8 12 37 60 



(12) (13) (14) (15) (16) (17) (18) (19) (20) 
27 16 50 21 270 393 1000 567482 2f — 1 = 
3 9 25 9 190 178 537 106493 



(21) 

10.00 

3.49 


(22) 
3i-| = 


(23) 
80836465 
49178036 


(24) 
81 
5f 


(25) 

27 

12-1 


(26) 
4 yds. 1 ft. 6 in. 
2 yds. 2 ft. 3 in. 



(27) 
5 yds. 1 ft. 4 in. 
2 yds. 2 ft. 8 in. 


(28) 
10 — 6.25 = 


(29) 
751 
52i 


(30) 
9.8063 — 9.019 



7.3 



(31) 


(32) 


(33) 


(34) 


(35) 


3.00081 = 


1912 6 mo. 8 da. 


5 2 


6i 


3i — 1 




1910 7 mo. 15 da. 


12 10 


21 





6 Measurements of Some Achievements in Arithmetic 

Series A 
MULTIPLICATION SCALE 

Name 

When is your next birthday? How old will you be?...., 

Are you a boy or girl? In what grade are you? 



(1) (2) (3) (4) (5) (6) (7) 
3X7= 5X1= 2X3= 4X8= 23 310 7X9 

3 4 



(8) (9) (10) (11) (12) (13) (14) (15) 

50 254 623 1036 5096 8754 _ 165 235 

3 6 7 8 6 8 40 23 



(16) (17) (18) (19) (20) (21) (22) 

7898 145 24 9.6 287 24 8 X 5f = 

9 206 234 4 .05 Ih 



(23) (24) (25) (26) (27) (28) (29) 

1JX8= 16 |X|= 9742 6.25 .0123 |X2 = 

21 59 3.2 9.8 



(30) (31) (32) {33) (34) 

2.49 12 15 6 dollars 49 cents 2| X 3§ = 1x1 = 

36 — X — = 8 
25 32 



(35) (36) (37) (38) (39) 

987f 3 ft. 5 in. 2i X 4§ X 1|= .0963i 8 ft. 9| in. 

25 5 .084 9 



The Arithmetic Scales and Their Uses 7 

Series A 
DIVISION SCALE 

Name '. 

When is your next birthday? How old will you be? 

Are you a boy or girl? In what grade are you? 



(1)_ (2^ (3) (4) (5) ( 6) 

316 91 27 41 28 115 9 1 36 3139 



(7) (8)_ (9)_ (10) (11) (12) 

4^2= 910 11 1 6X = 30 2 113 2 -=- 2 



(13) (14) (15) (162 ^^^^ 

4 1 24 lbs. 8 oz. 81 5856 i of 128 = 68 ] 2108 50 -^ 7 



(18) (19) (20) (21) (22) 

13 165065 248-^7= 2.1125.2 25 19750 2113.50 



(23) (24) (25) (26) 

23 1 469 75 ] 2250300 2400 ] 504000 12 ] 2.76 



(27) (28) (29) (30) 

I of 624= .003 1 .0936 3Jh-9= f-^5 = 

(31) (32) (33) 

5 3 9f H- 3f = 52 1 3756 

4*5 

(34) (35) (3_6) 

62.50 ^ U = 531 1 37722 9 ] 69 lbs. 9 oz. 



8 Measurements of Some Achievements in Arithmetic 

Series B 
ADDITION SCALE 

Name 

When is your next birthday? How old will you be? 

Are you a boy or girl? In what grade are you? 



(1) 


(2) 


(3) 


(5) 


(7) 


(10) 


2 


2 


17 


72 


3 + 1 = 


21 


3 


4 


2 


26 




33 


" 


3 








35 


(13) 


(14) 




(16) 


(19) 


(20) 


23 


25 + 42 = 




9 


$ .75 


$12.50 


25 






24 


1.25 " 


16.75 


16 






12 
15 
19 


.49 


15.75 












(21) 


(22) 




(23) 


(24) 


(30) 


$8.00 


547 




* + i = 


4.0125 


21 


5.75 


197 






1.5907 


6t 


2.33 


685 






4.10 


3f 


4.16 


678 






8.673 




.94 
6.32 


456 
393 




















525 
240 






















152 










(33) 


iS6) 






(38 ) 




.49 


2 yr. 5 mo. 




25.091 + 100.4 + 25 + 98.28 + 19.3614 = 


.28 


3 yr. 6 mo. 










.63 


4 yr. 9 mo. 










.95 


5 yr. 2 mo. 










1.69 


6 yr. 7 mo. 










.22 
.33 






















.36 












1.01 












.56 












.88 












.75 












.56 












1.10 












.18 












.56 













The Arithmetic Scales and Their Uses 

Series B 
SUBTRACTION SCALE 

Name 

When is your next birthday? How old will you be?. 

Are you a boy or girl? In what grade are you? 



(1) (3) (6) (7) 

8 2 11 13 

5 17 8 



(9) (13) (14) (17) 

78 16 50 393 

37 9 25 178 



(19) (20) (24) (25) 

567482 2f — 1 = Sf 27 

106493 5f 12f 



(27) (31) (35) 

5 yds. 1 ft. 4 in. 7.3 — 3.00081= 3| — If = 

2 yds. 2 ft. 8 in. 



lo Measurements of Some Achievements in Arithmetic 

Series B 
MULTIPLICATION SCALE 

Name 

When is your next birthday? How old will you be? 

Are you a boy or girl? In what grade are you? 



(1) (3) (4) (5) 

3X7= 2X3= 4X8= 23 

3 



(8) (9) (11) (12) 

50 254 1036 - 5096 

3 6 8 6 



(13) (16) (18) (20) 

8754 7898 24 287 

8 9 234 .05 



(24) (26) (27) (29) 

16 9742 6.25 I X 2 = 

21 59 3.2 



(33) (35) (37) (38) 

2| X 3|= 987i 2i X 4i X If = .09631 

25 .084 



The Arithmetic Scales and Their Uses ii 

Series B 
DIVISION SCALE 



Name 

When is yotir next birthday? How old will you be?. 

Are you a boy or girl? In what grade are you? 



(1)_ (2) (7) (8^ 

316 91 27 4-=-2= 910 

V 



( 11) (14) (15) (17) 

2 1 13 81 5856 \ of 128 = 50 -^ 7 



(19) (232 ^^^^ ^^^^ 

248 ^ 7 = 23 1 469 I of 624 = .003 ] .0936 



(30) (34) (36) 



1^5= 62.50^11= 9 169 lbs. 9 oz. 



12 Measurements of Some Achievements in Arithmetic 

over until they are told to do so. When all have their pencils 
in hand, say, " Turn your papers over and answer the questions 
at the top of the page." (The number of questions to be an- 
swered can be determined by the one giving the tests. It will 
take less time and cause less confusion if the one giving the 
tests will repeat the question and tell the children what to 
write. For example say, " The first line asks, ' What is your 
name?' Write your name," etc.) 

When all the questions have been answered repeat the follow- 
ing formula of specific directions. If you should happen to be 
giving the Addition test say, " Every problem on the sheet which 
I have given you is an addition problem, an " and problem." 
Work as many of these problems as you can and be sure that 
you get them right. Do all of your work on this sheet of paper 
and don't ask anybody any questions. Begin." 

For each scale in Series A, allow twenty minutes ; for each 
in Series B, allow ten minutes. It is important that the time be 
kept accurately and that all of the children quit work when the 
signal " Stop " is given. Most of the children will have finished 
before that time. Those who do not have done, in all proba- 
bility, all they can ; at least they have taken as much time as it 
takes the average class to complete the test. 

The only variation in procedure in giving any of the other 
tests is the substitution in the formula of specific directions of 
the expressions subtraction or " take away problems," multipli- 
cation or " times problems," and division or " into problems," 
for the expression addition or " and problems." The expres- 
sions "and," "take away," "times," and "into" problems are 
used so as to make clear to the children what process is to be 
involved. It is possible that teachers use these expressions in 
the lower grades instead of " addition, subtraction, multiplica- 
tion and division problems." There is a great variation in the 
names applied to the subtraction process. In giving the original 
tests it was necessary to find out how the teacher designated the 
process and then use her terminology. 

2. Directions for Scoring the Tests 

In scoring the tests the standard for marking a problem cor- 
rect is absolute accuracy, and, wherever possible, reduction to 



The Arithmetic Scales and Their Uses 13 

TABLE I : Answers to Problems 



PROBLEM 


ADDITION 


SUBTRACTION 


MULTIPLICATION 


DIVISION 


1 


5 


3 


21 


2 


2 


9 


6 


5 


3 


3 


19 


1 


6 


7 


4 


98 


6 


32 


5 


5 


98 





69 


4 


6 


97 


4 


1,240 


13 


7 


4 


5 


63 


2 


8 


8 


47 


150 





9 


87 


41 


1,524 


1 


10 


89 


3 


4,361 


5 


11 


108 


•16 


8,288 


6-1/2 not 6 + 1 


12 


59 


24 


30,576 




13 


64 


7 


70,032 


6 lbs. 2 oz. not 
6 2 


14 


67 


25 


6,600 


732 


15 


425 


12 


5,405 


32 


16 


79 


80 


71,082 


31 


17 


844 


215 


29,870 


7-1/7 not 7-f 1 


18 


10,966 


463 


5,616 


5,005 


19 


$2.49 


460,989 


38.4 


35-3/7 not 
35+3 


20 


$45.00 


1-3/4 


14.35 


12 


21 


$27.50 


6.51 


60 


390 


22 


3,873 


3 


46. 


6.75 


23 


2/3 


31,658,429 


10 


20-9/23; 20.3, 
not 20+9 


24 


18.3762 


3-1/8 


42 


30,004 


25 


2, not 16/8 nor 

2/1 
125, not 


14-3/8 


21/32 


210 


26 


1 yd. 2 ft. 3 in. 


574,778 


.23 




123-4/2 = 2 


not 63 in. 






27 


7/8 


2 yds. 1 ft. 8 in. 
not 81 in. 


20.000 


546 


28 


1 not 4/4 nor 
1/1 


3-3/4 or 3.75 


.12,054 


31.2 


29 


12-1/4 not 
11-3/4 = 
1-1/4 


23-1/2 not 
23-2/4 = 
1/2 


1/4 not 2/8 


7/18 


30 


12-5/8 not 
11-13/8 = 
1-5/8 


.7873 


89.64 


3/20 or .15 


31 


217.1413 


4.29919 


9/40 


2-1/12 


32 


1-1/2 not 6/4 


1 yr. 10 mo. 


$51.92 or 






nor 1-2/4 = 


23 da. 


51dol.92cts. 


2-17/30 




1/2 








33 


10.55 


13/60 


8-3/4 


72-3/13 or 

72.23 
50. 


34 


13/24 


3-1/4 not 


1/4 






3-2/8 = 1/4 






35 


10 ft. 8 in. or 


2-1/4 not 


24693-3/4 


71-7/177 or 




10-2/3 ft. 


2-2/8 == 1/4 




71.04 


36 


22 yrs. 5 mo. or 
22-5/12 yrs. 




17 ft. 1 in. 


7 lbs. 11-2/3 
oz.; 7 lbs. 
9 oz. 


37 


82-17/24 




15-3/16 




38 


268.1324 




.0080902-1/2 or 
.00809025 




39 






79 ft. 1-1/2 in. 





14 Measurements of Some Achievements in Arithmetic 

its lowest terms. If the results are to be comparable with the 
results and values established in these scales, only those answers 
should be accepted as correct which are found in Table I. These 
are the answers which were accepted in the original development 
of the scales. 

A few incorrect answers are also listed in order to offer less 
chance for variation in the scoring of the results. 

3. Directions for Determining the Class Score 

For the determination of the class score, two different methods 
have been derived. The first method was derived especially for 
use in Series A, where there is no definite attempt to place the 
problems on a linear scale with equal steps between them. By 
this method, after the problems have been marked as right or 
wrong, enter the results on a score sheet similar to the one given 
in Table 11. Thus a complete record of the particular problems 
solved by each child is obtained. 

To complete the class score, find the number of pupils in the 
class that solved each problem correctly. Divide the number 
by the total number in the class so as to get for each problem 
the per cent of the class that solved it correctly. Since, in the 
development of these scales, that problem which can be solved 
correctly by just 50 per cent of the class is taken as the best 
measure of the achievement of the class, select those five prob- 
lems which come nearest to being solved by just 50 per cent of 
the class.^ Table III gives the established value for each prob- 
lem in the different processes. From Table IV find the amount 
that must be added or subtracted to the values given in Table 
III for each of these selected problems to find just what difficulty 
a problem would need be in order that just 50 per cent of the 
class could solve it. Take the average of these five determina- 
tions and let it represent the class score. This means that a 
problem of that difficulty can be solved by just 50 per cent of 
the class in question. 

1 The work of scoring may be greatly economized by omitting the 
scoring and entering on the score sheet of the problems which will 
not figure in the determination of the 50 per cent right point. Thus 
in an eighth grade class the first twenty or more problems can most 
certainly be neglected. A little experience will teach the scorer what 
problems he needs to score for a given class. 



The Arithmetic Scales and Their Uses 



15 



TABLE II 
Sample Score Sheet 



PUPILS NAMES 


i 


2 


34 


5 


6 


7 


NO. OF PROBLEM 
a 9 10 1112 1314151617 1819 2021 22 23 


ETC 


































... 






































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































A 


















































—^""^ "^ "'**' \,.,f-^ 












- -J 








^^ 


























X 


r-M 





, ^_-^ ^* J— 


f 


"^ 


1 — I 


1 — 1 




r~i 




pi 


M 


r-1 


' 




—' 


' 




p-| 


r-n 


p> 




M 


_> 




n 


r~Ti 


Ho. Getting Each Problem 


















































$ Getting Each Problaa 












































J 







To illustrate the determination of the class score, the five 
problems in addition which came nearest to being solved by- 
just 50 per cent of the pupils in a certain third grade class of 
61 pupils were problems Nos. 14, 17, 16, 18 and 15. These 



l6 Measurements of Some Achievemewts in Arithmetic 

TABLE III 
Established Value of Each Problem in Each Scale 



NO. OF 










'ROBLEM 


ADDITION 


SUBTRACTION 


MULTIPLICATION 


DIVISION 


1 


1.23 


1.06 


.87 


1.57 


2 


1.40 


1.48 


1.05 


2.08 


3 


2.50 


1.50 


1.11 


2.18 


4 


2.61 


1.50 


1.58 


2.31 


5 


2.83 


1.70 


2.38 


2.40 


6 


3.21 


1.75 


2.62 


2.46 


7 


3.26 


2.18 


2.68 


2.56 


8 


3.35 


2.51 


2.71 


3.05 


9 


3.63 


2.57 


3.78 


3.16 


10 


3.78 


2.65 


3.79 


3.20 


11 


3.92 


2.88 


4.09 


3.49 


12 


4.18 


2.90 


4.26 


3.59 


13 


4.19 


2.96 


4.71 


3.96 


14 


4.85 


3.64 


4.72 


4.06 


15 


4.97 


3.70 


4.73 - 


4.60 


16 


5.52 


4.35 


5.05 


4.67 


17 


5.59 


4.41 


5.20 


4.98 


18 


5.73 


4.42 


5.24 


5.16 


19 


5.75 


5.18 


5.38 


5.26 


20 


6.10 


5.52 


5.63 


5.31 


21 


6.44 


5.70 


5.72 


5.36 


22 


6.79 


5.75 


5.83 


5.48 


23 


7.11 


5.76 


5.83 


5.56 


24 


7.43 


5.91 


5.89 


5.58 


25 


7.47 


6.77 


6.29 


5.78 


26 


7.61 


7.07 


6.30 


5.91 


27 


7.62 


7.21 


6.58 


6.04 


28 


7.67 


7.38 


6.85 


6.43 


29 


7.71 


7.41 


6.97 


6.76 


30 


7.71 


7.41 


7.00 


6.83 


31 


7.97 


7.49 


7.07 


6.87 


32 


8.04 


7.52 


7.07 


6.88 


33 


8.18 


7.69 


7.29 


7.22 


34 


8.22 


7.72 


7.50 


7.24 


35 


8.58 


7.84 


7.65 


8.17 


36 


8.67 




7.66 


8.23 


37 


8.67 




8.02 




38 


9.19 




8.53 




39 






8.61 





problems were thus solved correctly by 54, 54, 48, 48 and 68 
per cent of the class, respectively. Table IV tells how much to 
add to or subtract from the established value given in Table III 



The Arithmetic Scales and Their Uses 



17 



i 1 f 



o 

>. 

o 

'-3 



i8 Measurements of Some Achievements in Arithmetic 



COMPOSITION OF SCALES IN "SERIES B" 
Addition Subtraction Multiplication Division 



NO. OF 




NO. OF 




NO. OF 




NO. OF 




PROBLEM 


VALUE 


PROBLEM 


VALUE 


PROBLEM 


VALUE 


PROBLEM 


VALUE 


1 


1.23 


1 


1.06 


1 


.87 


1 


1.57 


2 


1.40 


3 


1.50 


3 


1.11 


2 


2.08 


3 


2.50 


6 


1.75 


4 


1.58 


7 


2.56 


5 


2.83 


7 


2.18 


5 


2.38 


8 


3.05 


7 


3.26 


9 


2.57 


8 


2.71 


11 


3.49 


10 


3.78 


13 


2.96 


9 


3.78 


14 


4.06 


13 


4.19 


14 


3.64 


11 


4.09 


15 


4.60 


14 


4.85 


17 


4.41 


12 


4.26 


17 


4.98 


16 


5.52 


19 


5.18 


13 


4.71 


19 


5.26 


19 


5.75 


20 


5.52 


16 


5.05 


23 


5.57 


20 


6.10 


24 


5.91 


18 


5.24 


27 


6.04 


21 


6.44 


25 


6.77 


20 


5.63 


28 


6.43 


22 


6.79 


27 


7.21 


24 


5.89 


30 


6.83 


23 


7.11 


31 


7.49 


26 


6.30 


34 


7.24 


24 


7.43 


35 


7.84 


27 


6.58 


36 


8.23 


30 


.7.71 






29 


6.97 






33 


8.18 






33 


7.29 






36 


8.67 






35 


7.65 






38 


9.19 






37 
38 


8.02 
8.53 







for each of the problems in order to estimate the value of a prob- 
lem that vv^ould be solved correctly by just 50 per cent of the class. 
Thus 

For 54 per cent add .15 to 4.85 = 5.00 



54 per cent " .15 

68 per cent " .70 

48 per cent subtract 

48 per cent " 



5-59 =5-74 

" 4-97 =5-67 

.07 from 5.52 = 5.45 

•07 " 5-73 = 5-66 
Average 5.50 



The average of these 5 determinations (5.50) represents better 
than either single measurement the degree of difficulty that a 
problem must have in order that just 50 per cent of this class 
can solve it correctly. The class score for any other class can 
be computed in a similar manner. 

The second method for the determination of the class score 
was derived especially for Series B wrhere there wsls a definite 
attempt to place the problems on a linear scale with equal steps 
between them. This method introduces a certain amount of 
error, but for all practical purposes it is a satisfactory measure. 
By this method the median number of problems solved cor- 



The Arithmetic Scales and Their Uses 19 

TABLE IV 

For Use in Estimating the Degree of Difficulty Required 

IN A Problem so That Just 50 Per Cent of the Class 

CAN Solve it Correctly 

subtract add 



10% 


1.90 


11 


1.82 


12 


1.74 


13 


1.67 


14 


1.60 


15 


1.54 


16 


1.48 


17 


1.42 


18 


1.36 


19 


1.30 


20 


1.25 


21 


1.20 


22 


1.15 


23 


1.10 


24 


1.05 


25 


1.00 


26 


.95 


27 


.91 


28 


.86 


29 


.82 


30 


.78 


31 


.74 


32 


.70 


33 


.65 


34 


.61 


35 


.57 


36 


.53 


37 


.49 


38 


.45 


39 


.41 


40 


.38 


41 


.34 


42 


.30 


43 


.26 


44 


.22 


45 


.19 


46 


.15 


47 


.11 


48 


.07 


49 


.03 



50% 


0.00 


51 


.03 


52 


.07 


53 


.11 


54 


.15 


55 


.19 


56 


.22 


57 


.26 


58 


.30 


59 


.34 


60 


.38 


61 


.41 


62 


.45 


63 


.49 


64 


.53 


65 


.57 


66 


.61 


67 


.65 


68 


.70 


69 


.74 


70 


.78 


71 


.82 


72 


.86 


73 


.91 


74 


.95 


75 


1.00 


76 


1.05 


77 


1.10 


78 


1.15 


79 


1.20 


80 


1.25 


81 


1.30 


82 


1.36 


83 


1.42 


84 


1.48 


85 


1.54 


86 


1.60 


87 


1.67 


88 


1.74 


89 


1.82 


90 


1.90 



rectly is taken as the ;measure of the achievement of any class. 
By the median number of problems solved is meant such a 
number of problems that there are just as many pupils who 
solve a greater number as there are those who solve a less 
number. 

In order to determine the median point of the achievement 
of the class, it is necessary to make a distribution table, show- 



20 Measurements of Some Achievem^ents in Arithmetic 

ing the number of pupils who were unable to solve a single 
problem correctly, the number who solved one problem, two 
problems, three problems, etc. As examples of this sort of 
distribution we may take the following: 

TABLE V 
Number of Times Each Addition Problem was Solved Correctly 








1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


Class I . . . 


1 





2 


3 





4 


1 


2 


3 


5 


6 


11 


1 


5 


2 


4 


1 


1 




Class II . . . 


3 


3 


4 


4 


7 


5 


3 


4 


3 




1 


1 
















Class III ... 




1 


2 


4 


6 


7 


4 





4 


5 


5 


4 


3 


2 


1 











According to these distributions 52 pupils are in Class I, 
37 pupils in Class II, and 48 pupils in Class III. Now let us 
proceed to find the median achievement for each of these class 
distributions. 

Since there are 52 individuals in Class I the median point 
evidently falls between the achievements of the 26th and 27th 
pupils. Let us begin with the individual who was unable to 
solve a single problem correctly and count the two individuals 
who solved two problems, the three who solved three problems, 
and so on till we come to the steps that includes the 26th indi- 
vidual. Now if we are to indicate the exact point in the achieve- 
ment of the pupils where there are just as many pupils who 
solve a greater number of problems as where there are those 
who solve a less number, it is necessary to count 5 of the 6 
individuals who solved 10 problems correctly. Thus on the 
assumption that the individuals are distributed over any step 
at equal distances from one another, the median point is 5/6 of 
the distance through this step. Hence: the median achieve- 
ment of this class, i.e., the median number of problems solved, 
is 10.8 problems correctly solved. 

Similarly there are 37 pupils in Class II. The middle case 
is the 19th pupil, who is the fifth pupil "in step 4. There are 
18 pupils who solve a greater number of problems and 18 who 
solve a less pumber of problems. Thus the exact median point 
in the achievement of the class lies in the middle of that frac- 
tion of a step assigned to the 19th pupil. Thus the median 



The Arithmetic Scales and Their Uses 21 

point is ^Y^ of the distance through the 4th step. Hence the 
median achievement for this class is 4.6 problems solved correctly. 
The distribution for Class III represents a peculiar difficulty 
in the calculation of its median. There are 48 pupils in this 
class and evidently the median point falls between the 24th 
and the 25th individual. However, it happens that 24 of the 
pupils solve more than seven problems and 24 of them solve 
less than seven. Probably the wisest assumption to make is 
that the 4 pupils on step 6 take up all of that step and the 4 
pupils on step 8 take up all of that. If this is assumed, then 
the median falls on step 7, probably at 7.5 since any given dis- 
tance on a scale is best represented by its middle point. Thus 
the median achievement for the Class III is 7.5 problems solved 
correctly. By similar computations the medians of any dis- 
tribution can be obtained. By the comparison of the medians 
thus determined, we get a very satisfactory measure of the 
achievement of any class on the basis of the total number of 
problems correctly solved. 

4. Tentative Standards of Achievement 

While these new scales have not been used in measuring suf- 
ficient numbers of children to warrant the establishment of 
definite standards of achievement, it was thought well to indicate 
some tentative standards. These tentative standards have been 
derived from the actual achievements of the children tested 
with the preliminary tests. The fact that these tests were given 
during the first part of the school year should be kept in mind 
when comparison is made with tentative standards shown in 
Tables VI and VII. 

Table VI contains the tentative standards for Series A. 

TABLE VI 
Tentative Standards of Achievement for Series A 

SUBTRACTION MtlLTIPLICATION DIVISION 

1.44 

2.96 1.89 2.54 

4.22 4.05 3.21 

5.47 5.53 4.94 

6.46 6.72 5.87 

7.31 7.26 6.59 

7.64 7.93 7.16 



grade 


ADDITION 


II 


3.12 


III 


4.99 


IV 


6.11 


V 


6.99 


VI 


7.95 


VII 


8.65 


VIII 


9.01 



22 Measurements of Some Achievements in Arithmetic 

These standards were derived according to the first method 
given for the determination of the class score. They are based 
upon the degree of difficulty which the problems must possess 
in order that just 50 per cent of the class can solve them. Thus, 
if a problem in addition has 3.12 units of difficulty it will be 
solved by 50 per cent of the second grade; if it has 4.99 |Units 
of difficulty it will be solved by 50 per cent of the third grade, etc. 

Table VII contains the tentative standards for Series B. 

TABLE VII 
Tentative Standards of Achievement for Series B 

SUBTRACTION Mt 

3 

6 

8 
10 
12 
13 
14.5 

These standards have been derived according to the second 
method for determining the class score. They are based upon 
the total number of problems that were correctly solved in each 
grade. Thus in the second grade in addition, the median achieve- 
ment was 4,5 problems, in the third grade, 9 problems correctly 
solved, etc. • ' 



3RADE 


ADDITION 


II 


4.5 


III 


9 


IV 


11 


V 


14 


VI 


16 


VII 


18 


VIII 


18.5 



■LIGATION 


DIVISION 


3.5 


3 


7 


5 


11 


7 


15 


10 


17 


13 


18 


14 



Section III. THE VALUE AND USES OF THE SCALES 

1. The scales themselves contain 148 problems which in- 
volve many of the fundamental principles of arithmetic. A 
child who understands and can solve all of these problems 
correctly probably knows more arithmetic than the average 
eighth grade child. 

2. These scales are useful in that the value of each problem 
is known, and from these values the value of other problems 
can easily be determined. 

3. The scales are useful in measuring the achievements of 
any class or of a whole school system. Since all the pupils 
in all the grades are measured by the same scales, the amount 
of progress from grade to grade can be definitely determined. 
Comparisons can be made with similar grades in other build- 
ings or school systems. If the measurements show, for instance, 
that a certain sixth grade class is unable to solve a greater num- 
ber of problems correctly than a fifth grade class in the same 
school system, the cause of this condition should be investigated. 
In such ways the tests should prove useful to those in charge of 
school systems. 

4. Perhaps the most valuable use of the scales lies in the 
diagnosing power of the class mistakes. The writer was con- 
vinced during the process of scoring these test papers, nearly 
20,000 in all, that the mistakes of a class tend to be grouped 
around some central tendency. The great variety of the prob- 
lems in these scales and the fact that the problems in each of 
the various operations proceed from the simplest to the more 
difficult problems aid greatly in locating the weaknesses of the 
class. If a large number in a class fail to invert the divisor in 
the problems in division of fractions, or if a large number in 
a class fail to locate the decimal point properly in the problems 
in multiplication of decimal fractions, a teacher should know 
immediately that these classes need more practice in these par- 
ticular processes. In a like manner, by locating the particular 
types of problems missed, one should be able to direct the work 
of a class more intelligently. 

23 



Section IV. LIMITATIONS OF THE SCALES 

1. It is possible that with a greater number and variety of 
pupils the value of some of the problems might be somewhat 
changed. However, the children tested were from widely sepa- 
rated districts in Indiana, New Jersey, Connecticut, and New 
York. They represent children from many classes of society and 
from many nationalities. Moreover, much variation existed in 
the methods of teaching and in the school room practices. Thus 
the writer believes the values established are well founded. 

2. On the scales as now presented the value of some of the 
problems may be slightly altered due to the fact that they are 
located in different positions from those in which they were 
located on the preliminary lists of problems. The exact amount 
of this alteration can be determined only by further testing 
with the scales. 

3. The scales as now presented might be slightly bettered if 
two or three more difficult problems were added to each of them. 
The scales probably would be bettered if problems could be 
found of such difficulty as to make the steps between them 
of exactly equal distance. However, for practical purposes, 
the effects of these two defects can be disregarded. 

4. The value of these scales may be somewhat affected by 
their more extended use. As teachers become more acquainted 
with them, they may drill especially upon them. Therefore, 
it would be much better if several series of such scales of the 
same difficulty as these should be developed. 1 

5. These scales are not intended to give a definite measure 
of an individual child. But, if we can measure approximately 
how difficult a problem a child can solve and then supplement 
this problem with a large list of problems similar in nature 
and in difficulty, we can get a fairly accurate measure of the 
achievement of the child. 

6. The relative difficulty of these problems was determined 
from the achievements of school children in grades' two to eight 
inclusive. It is probable that for adults and teachers the rank- 
ing would be in a different order. Only further testing can 
substantiate this point. 

24 



PART II 

Section L DERIVATION OF THE SCALE 

I. History of the Scale 

The completed scales as shown in Part I of this monograph 
have been developed from about 20,000 test sheets. The first 
preliminary series of tests were given to a number of pupils 
in the public schools of Indiana and New Jersey. The pre- 
liminary series of tests consisted of a sheet of problems in 
addition and likewise one in subtraction, multiplication, and 
division. In constructing these preliminary lists there was a 
definite attempt to select problems of as great a variety as the 
fundamental processes would permit. There was also an at- 
tempt to begin the series in each process with the easiest 
problem that could be found and then gradually to increase 
the difficulty of each succeeding problem until the last ones in 
the series would be correctly solved by only a small percentage 
of the pupils in the eighth grade. By the selection of problems 
of such varied types and by giving the same lists of problems 
to pupils in all grades, it was thought that the diagnosing power 
of the lists would be greater and that the amount and the 
nature of the progress of one grade over another could best 
be determined. 

The preliminary lists of problems in addition were given to 
908 pupils, in subtraction to 916 pupils, in multiplication to 
868 pupils, and in division to 696 pupils. The results of these 
preliminary lists showed that some of the problems were poorly 
selected and that they should be discarded. When the prob- 
lems were ranked according to the total percentage of pupils 
solving them correctly, the results showed large gaps existing 
between the problems in particular portions of the series. 

Guided by the results of these preliminary lists new lists 
were constructed. Only those problems of the original lists 
were chosen which were solved by a gradually increasing per- 
centage of the pupils as one proceeded from the lower to the 
higher grades. If a problem were solved by a higher per- 
centage of the pupils in the lower grades than in the higher 

25 



26 Measurements of Some Achievements in Arithmetic 

grades it was rejected. Wherever there tended to be too large 
a step between two consecutive problems in the original series 
an attempt was made to interpose two or three problems of 
intermediate difficulty, ; 

From the last week in October till the end of the second 
week in December, 191 5, pupils were tested with these new 
lists of problems. These pupils were from seven different school 
systems located in Indiana, New Jersey, Connecticut, and New 
York. The addition problems were given to 4,489 pupils, the 
subtraction to 4,423 pupils, the multiplication to 3,922 pupils, 
and the division to 3,660 pupils. These pupils were distributed 
fairly equally from the second to the eighth grades inclusive. 

All of the tests were given by the writer himself with the 
exception of those given to the pupils in two small school sys- 
tems in Indiana.^ The tests were given and the results scored 
according to the instructions given for administering the tests 
in Part I of this monograph with the one exception that no time 
limit for the solution of the problems was used. It was felt 
to be highly important, if the difficulty of each problem was 
to be firmly established, that each child should have a chance 
to solve each problem. 

All of the tests were scored by the writer himself and thus 
the personal element in scoring was reduced to a minimum. 
The standard for marking a problem right or wrong as pre- 
sented in Part I of this monograph was arbitrarily adopted. 
It was decided that a problem to be marked correct must be 
absolutely accurate and, wherever possible, reduced to its 
lowest terms. Otherwise, the problem was marked wrong. How- 
ever, before adopting this arbitrary standard an effort was made 
to gain from teachers and supervisors of arithmetic the stand- 
ards by which they marked a problem right or wrong. It was 
almost unanimously agreed that a problem must be absolutely 
accurate and reduced to its lowest terms. Thus the arbitrary 
standard adopted by the writer is in accordance with the best 
practice exercised in the teaching of arithmetic. 

The results of these tests were recorded in two ways: 

I. The pupils were distributed according to the number of 

1 Those giving the tests in these two systems were men who have 
had experience in giving tests and who could be trusted to carry out 
the writer's directions. 



Derivation of the Scale 27 

TABLE VIII 
Distribution According to the Number of Addition Problems Solved 

grade grade grade grade grade grade grade 

ii iii iv v vi vii viii 

38 3 21 41 

37 15 37 33 

36 30 82 55 

35 2 37 96 72 

34 4 45 91 70 

33 . 1 51 90 76 

32 8 34 75 46 

31 13 45 83 45 

30 1 13 45 49 27 

29 26 51 57 20 

28 2 35 33 53 18 

27 1 32 36 48 19 

26 1 2 46 37 34 10 

25 11 40 34 34 4 

24 ' 5 54 37 16 4 

23 3 33 75 29 27 1 

22 6 47 64 25 8 2 

21 1 11 42 77 15 7 1 

20 10 54 54 15 4 

19 26 65 49 6 3 

18 43 56 43 5 2 

17 4 47 75 28 3 

16 3 64 72 10 2 

15 7 70 42 7 

14 7 54 24 3 

13 13 44 14 

12 10 40 18 1 

11 43 39 16 1 

10 31 33 7 

9 46 35 5 

8 38 23 2 

7 35 16 1 

6 36 10 2 

5 69 14 1 1 

4 48 8 2 

3 43 4 1 

2 17 4 

1 13 6 1 

25 4 

No. Tested. . 489 615 602 687 633 917 544 

Median 6.819 14.509 18.321 23.073 29.774 32.446 33.987 

25 per cent.. 4.505 10.902 16.201 20.532 25.625 28.872 31.667 

75 " " . . 9.929 16.894 20.694 26.206 33.446 35.070 35.903 

Quartne 2.712 2.996 2.247 2.837 3.910 3.099 2.118 



28 Measurements of Some Achievements in Arithmetic 

TABLE IX 

Number in Each Grade that Solved Each Problem in Addition 
Correctly 

PROBLEM grade GRADE GRADE GRADE GRADE GRADE GRADE 
NO. II III IV V VI VII VIII 



1 


388 


456 


499 


654 


622 


896 


541 


2 


433 


582 


595 


681 


630 


913 


542 


3 


392 


593 


595 


680 


626 


911 


539 


4 


326 


468 


521 


659 


614 


901 


540 


5 


323 


501 


554 


673 


629 


914 


544 


6 


279 


530 


565 


668 


628 


911 


542 


7 


259 


538 


565 


679 


631 


915 


542 


8 


220 


474 


542 


665 


624 


911 


544 


9 


165 


530 


568 


667 


613 


880 


522 


10 


152 


531 


570 


663 


623 


895 


539 


11 


190 


543 


577 


663 


608 


886 


535 


12 


52 


399 


541 


657 


620 


896 


537 


13 


32 


229 


373 


627 


622 


901 


537 


14 


37 


405 


541 


664 


627 


900- 


534 


15 


23 


328 


499 


627 


602 


876 


530 


16 


6 


238 


387 


567 


533 


806 


500 


17 


22 


288 


431 


551 


500 


787 


475 


18 


8 


208 


386 


539 


511 


801 


505 


19 


1 


92 


246 


399 


436 


662 


457 


20 


1 


87 


307 


555 


586 


883 


528 


21 


1 


71 


276 


498 


564 


839 


498 


22 




49 


204 


441 


528 


814 


489 


23 




4 


34 


308 


490 


771 


500 


24 







4 


99 


296 


521 


385 


25 




3 


14 


213 


397 


651 


457 


26 




2 


11 


192 


423 


682 


483 


27 







10 


178 


369 


678 


470 


28 







14 


166 


414 


693 


448 


29 







8 


131 


300 


591 


409 


30 




2 


33 


157 


403 


684 


462 


31 




3 


34 


164 


317 


490 


344 


32 







4 


157 


373 


674 


421 


33 







3 


57 


235 


461 


290 


34 




15 


128 


271 


338 


684 


432 


35 




4 


57 


169 


318 


558 


392 


36 







2 


40 


179 


537 


354 


37 







1 


20 


176 


529 


359 


38 







1 


9 


155 


240 


274 



No. Tested. 489 615 602 687 633 917 544 



Derivation of the Scale 29 

problems solved correctly. Table VIII represents the distribu- 
tion for the problems in addition. Beginning at the lower left- 
hand corner, Table VIII shows that 25 out of 489 pupils in 
the second grade, and 4 out of 615 pupils in the third grade 
were unable to solve a single problem, etc. This table also 
shows the median achievement of each grade distribution. The 
median achievement of a class is such a number of problems 
correctly solved that there are just as many pupils who solve 
a greater number of problems as there are those who solve 
a less number. This table shows the range in the number of 
problems correctly solved that will include the middle 50 per 
cent of the pupils. It also shows the variability in terms of the 
quartile, or, as it is sometimes designated, the " semi-inter- 
quartile range." 

2. The results were tabulated in another method so as to 
record the number of pupils who solved each individual problem 
correctly. Thus Table IX shows that 388 out of 489 pupils in 
the second grade solved problem No. i ; 433 pupils solved 
problem No. 2, etc. From these two crude summaries given 
in Tables VIII and IX the addition scales have been developed.^ 

2. P.E. AS A Unit of Measure 
It may be said that we have always measured pupils in the 
fundamental operations of arithmetic. It may be said that 
schools and school systems have likewise been measured. No 
doubt this is true. Whenever a teacher says that one boy is 
better in addition than another boy, in a certain sense, she 
measures him. Whenever we compare one individual with 
another individual, one quality with another quality, or one 
class with another class, we are measuring. Such standards of 
measurements as these are no doubt inaccurate and changeable. 
Whenever a teacher measures a class by means of an examina- 
tion she tends to have a more constant and more objective meas- 
urement. The relation of the different questions of the exam- 

1 Similar tables for the problems in subtraction, multiplication, and 
division will be found at the end of Part II. In the discussion of 
the derivation of the scales I shall show in detail the method by 
which the scale in addition was developed, and shall not discuss the 
other processes. However, I shall include the final values of each 
problem in each of the other processes and the most important 
tables of crude data from which the established values were deter- 
mined. 



30 Measurements of Some Achievements in Arithmetic 

ination to one another, however, is unknown. All the questions 
may be of equal difficulty, or one may be several times as diffi- 
cult as another. The chief value of a scale as a means of meas- 
urement is ^that it is made up of a number of distinct units 
whose value is known and remains constant. Such a scale can 
be used by different people in making similar measurement 
and the results will be comparable. On the linear rule the unit 
of measurement is the inch or centimeter; on the thermometer, 
the degree. Everyone knows what is meant when we speak 
of an inch, a degree, or any fractional part thereof. These 
amounts are very definite and always have the same meaning. 
Moreover almost any one can make reliable measurements with 
a rule or with a thermometer. 

In the building of these arithmetic scales there has been a 
definite attempt to approximate as closely as possible the accur- 
acy and the constancy of the ruler or the thermometer. The 
difficulty of each problem has been established and its position 
above a selected zero point determined. The problems have all 
been placed in their relative positions on a projected linear scale. 
In the scales of Series B a definite attempt has been made to 
select problems with equal amounts of difficulty between them. 
The unit of measure of difficulty on these arithmetic scales, 
^hich corresponds to the inch on the ruler or to the degree 
on the thermometer, is what is called in statistical terms the 
Median Deviation or Probable Error. (P.E.) 

Before taking up the significance of the median deviation let 
us discuss the normal surface of frequency. In the construc- 
tion of these scales, it has been assumed that achievement in 
the solution of problems in the fundamental processes is dis- 
tributed according to the normal surface of frequency. Fur- 
thermore it has been assumed that the variability of any grade 
from the second to the eighth is equal to that of any other. 

These assumptions are based upon the well-established principle 
that intellectual abilities are distributed in the same way as are 
physical traits. If we should arrange one thousand men, selected 
at random, in a row according to their height, we should find 
a very large group of men in the center who are about medium 
height. On one end of the row would be a few very short men 
and on the other end would be a few very tall men. Likewise 



Derivation of the Scale 



31 



if we assume that achievement in the solution of problems in 
the fundamental processes in any grade is distributed normally, 
then we should expect to find a large number of the class solv- 
ing about the same number of problems ; furthermore we should 
expect to find a few dull pupils who can solve but just a few 
problems and a few bright pupils who can solve more than the 
average number of problems. The so-called normal curve illus- 
trating such a distribution is reproduced in Fig. i. The properties 
of the normal curve have been most accurately determined. 
Let us assume that Fig. i represents the achievement in the 
solution of problems among a large number of third grade pupils. 




very few few average many very many 

Fig. I. Normal surface of frequency showing the distribution of 
achievement in the solution of problems. 



The space enclosed between the curve and the base-line repre- 
sents all of the pupils arranged according to the number of 
problems solved. The height of the curve above the base-line 
indicates the number of pupils in the class solving the relative 
number of problems shown on the base-line. Each pupil is 
represented by an equal amount of the enclosed area. Thus, 
at the extreme left the curve is very near the base, which indi- 
cates the small number of pupils who were able to solve only a 
very few problems. In the middle the curve is distant from 
the base-line representing the large number of pupils who solved 
an average number of problems; at the extreme right the curve 
is ,very near the base, which indicates the small number of 
pupils who are able to solve many more than the average num- 
ber of problems. 

If our assumption with regard to the achievement in the solu- 
tion of problems is true, then the graphic representations of the 



32 Measurements of Some Achievements in Arithmetic 

tables of distribution according to the number of problems solved 
must be similar to Fig. i. 

Figs. 2 to 8 inclusive represent graphically the distribution 
of the achievement in the solution of the addition problems 
throughout the various grades. These figures on the whole 
correspond fairly well to the normal curve of distribution. It 
will be seen that in the second grade distribution the curve 
is somewhat skewed to the left. This is probably due to the fact 
that a great number of the teachers were just beginning to teach 
the fundamental operations to their classes. It will also be seen 
that the distributions for grades seven and eight are skewed 
somewhat to the right. This indicates the need for one or two 
more difficult problems at the end of the addition series. It 
will be noted from the distribution tables in the back of this 
monograph that the distributions for the other processes con- 
form to the normal curve better than the foregoing figures. 



V 



o=Median Score. 



Olii'iSiyi 1 /a II It IS If I sit 17 H 11 ZOZt itZittzfti, n Zizi 30 31 3Z333-(3S3i9T3» 

Fig. 2. Distribution according to the number of Addition Problems 
solved in Second Grade. 




^ 



'i n It II " i' « iJ M «-J- :4 K7 Z-lf Zl 3I> il 52 53 3¥ 3i Jt 57 jy 



Fig. 3. Distribution according to the number of Addition Problems 
solved in Third Grade. 



Derivation of the Scale 



33 




£ 3 "J £ (, 7 



i If If IS It n IS li ZO Zl ZZ 13 2f 25^4 ZJlsCliO 31 3C 13 3t 3S 3i JJ it 



Fig. 4. 



Distribution according to the number of Addition Problems 
solved in Fourth Grade. 



-^ 



Ln 



^ 




01 Z. 9 If S t T t f l» IJ U 13 Jif If H If It II to II 21 13 It XS t( 17 II I* 3» 5> 3Z 33 SH 3! Si JJ 3f 



Fig. 5- 



Distribution according to the number of Addition Problems 
solved in Fifth Grade. 




7 It If ZO tl Tt 13 t4 tSti Z7 t.1 tf so ) I iX 33 W 3r 3k tT3t 



Distribution according to the number of Addition Problems 
solved in Sixth Grade. 



34 Measurements of Some Achievements in Arithmetic 



nJ 



^ 



r 

u 



7 t 1 lO // /( , 



(1- IS /' n It II *" ii 



r«4 17 t» Z1 3i> J/ 



SJ 3* J5 3»S7»f 



Fig. 7. Distribution according to the number of Addition Problems 
solved in Seventh Grade. 




Fig. 8. Distribution according to the number of Addition Problems 
solved in Eighth Grade. 

Having examined the normal curve of distribution, let us 
define the Median Deviation or Probable Error, which has been 
used as the unit of measure in the construction of these arith- 
metic scales. Let us draw a perpendicular to the base of the 
surface of frequency so that fifty per cent of all the cases lie 
on one side of the perpendicular and fifty per cent on the other 
side. The point where the perpendicular cuts the base is the 
median point. To the left of the median point, draw a perpen- 
dicular / a so that just 25 per cent of the cases lie between it 
and the median perpendicular. Draw a similar perpendicular 
d c to the right of the median point. The area a c d f cut ofif 
by these perpendiculars contains the middle 50 per cent of all 
the cases. The distance a m or m c on the base-line of the 



Derivation of the Scale 



35 




Fig. 9. Normal Surface of Distribution showing the Median and P.E. 
distance at each side of the Median Point. 

surface of frequency is the Median Deviation or the Probable 
Error. The Probable Error or P.E., as it will be called through- 
out this monograph, is thus the distance along the base-line 
of a surface of distribution from the median point to the per- 
pendicular on either side of the median which cuts ofif 25 per 
cent of the cases. 

Furthermore, it has been established that 2 P.E. is the dis- 
tance from the median point to the perpendicular on either 
side of the median which cuts ofif 41.13 per cent of the cases; 
3 P.E., the distance which cuts off 47.85 per cent of the cases ; 
and 4 P.E., the distance which cuts off 49.65 per cent of the 
cases. Theoretically the curve and the base-line never meet but 
continually approach one another as the distance from the 
median point increases. For the purposes of this study we may 
consider that they meet at a distance of 4.6 P.E. from the 
median point, for a perpendicular erected here on either side 
of the median cuts off but o.i per cent of all the cases. These 
facts enable us to locate each problem in its proper position 
on the base of any grade distribution. 




-4 P.E. -3 P.E. -2 P.E. -I P.E. M. I P.E. 2 P.E. 3 P.E. 4 P.E. 

Fig. 10. Normal Surface of Frequency showing P.E. distances from 

the Median Point. 



36 Measurements of Some Achievements in Arithmetic 

3. Scaling the Problems in x\ddition for Each Grade 

Since we have assumed that achievement in the solution of 
problems in the fundamental processes is distributed according 
to the normal surface of frequency and since we have adopted 
the P.E. of a grade distribution as the unit of measurement, it 
is an easy matter to locate each problem on the base-line of 
each grade distribution. It is evident that a problem which is 
solved by exactly 50 per cent of pupils in any class represents 
the median achievement of the class and that it would be located 
at the median point of the base-line. By definition, P.E. is the 
distance along the base-line from the median point to the per- 
pendicular on either side of the median which cuts off 25 per 
cent of the cases. Evidently then a problem that is solved by 
75 per cent of the pupils would be i P.E. too easy to represent 
the median achievement of the class and would be located at 
-I P.E. distance from the median point. Likewise a problem 
that is solved by only 25 per cent of the pupils is too difficult to 
represent the median achievement and would be located at -f i 
P.E. distance from the median point. Thus, if we know what 
per cent of a class solved any problem, it is easy to find the 
deviation of this per cent from 50 per cent or the median achieve- 
ment of the class. If this per cent of deviation from the median 
achievement is known in terms of P.E., we can locate any 
problem with reference to the median of that distribution. Table 

X gives the P.E. value for each tenth of a per cent deviation 
from the median point of a normal distribution (i.e., deviation 
of 0.0 per cent to 49.9 per cent above or below the median).^ 

Table IX previously given (page 28) shows the number in 
each grade that solved each problem in addition correctly. Table 

XI shows these numbers reduced into terms of per cents. Thus 
in the second grade 79.4 per cent of the pupils solved problem 
No. I ; 88.6 per cent solved problem No. 2, etc. 

Table XII shows the difference between 50 per cent (the 
median achievement) and the per cents given in Table XI. 
Table XIII shows the P.E. values for the differences given in 
Table XII. These P.E. values represent the position of each 
problem on the base-line of the grade distribution with reference 

1 Table X is taken directly from B. R. Buckingham's Spelling 
Ability (Table XLVII). It is a modification of the table given in 
E. L. Thorndike's Mental and Social Measurements (page 200). 



Derivation of the Scale 37 

TABLE X 

P.E. Values Corresponding to Given Per Cents of the Normal 
Surface of Frequency, Per Cents Being Taken from the Median 

% .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 








.000 


.004 


.007 


.011 


.015 


.019 


.022 


.026 


.030 


.033 


1 




.037 


.041 


.044 


.048 


.052 


.056 


.059 


.063 


.067 


.071 


2 




.074 


.078 


.082 


.085 


.089 


.093 


.097 


.100 


.104 


.108 


3 




.112 


.115 


.119 


.123 


.127 


.130 


.134 


.138 


.141 


.145 


4 




.149 


.153 


.156 


.160 


.164 


.168 


.172 


.175 


.179 


.183 


5 




.187 


.190 


.194 


.198 


.201 


.205 


.209 


.213 


.216 


.220 


6 




.224 


.228 


.231 


.235 


.239 


.243 


.246 


.250 


.254 


.258 


7 




.261 


.265 


.269 


.273 


.277 


.280 


.284 


.288 


.292 


.296 


8 




.299 


.303 


.307 


.311 


.315 


.318 


.322 


.326 


.330 


.334 


9 




.337 


.341 


.345 


.349 


.353 


.357 


.360 


.364 


.368 


.372 


10 




.376 


.380 


.383 


.387 


.391 


.395 


.399 


.403 


.407 


.410 


11 




.414 


.418 


.422 


.426 


.430 


.434 


.437 


.441 


.445 


.449 


12 




.453 


.457 


.461 


.464 


.468 


.472 


.476 


.480 


.484 


.489 


13 




.492 


.496 


.500 


.504 


.508 


.512 


.516 


.519 


.523 


.527 


14 




.531 


.535 


.539 


.543 


.547 


.551 


.555 


.559 


.563 


.567 


15 




.571 


.575 


.579 


.583 


.588 


.592 


.596 


.600 


.603 


.608 


16 




.612 


.616 


.620 


.624 


.628 


.632 


.636 


.640 


.644 


.648 


17 




.652 


.656 


.660 


.665 


.669 


.673 


.677 


.681 


.685 


.689 


18 




.693 


.698 


.702 


.706 


.710 


.714 


.719 


.723 


.727 


.731 


19 




.735 


.740 


.744 


.748 


.752 


.756 


.761 


.765 


.769 


.773 


20 




.778 


.782 


.786 


.790 


.795 


.799 


.803 


.807 


.812 


.816 


21 




.820 


.825 


.829 


.834 


.838 


.842 


.847 


.851 


.855 


.860 


22 




.864 


.869 


.873 


.878 


.882 


.886 


.891 


.895 


.900 


.904 


23 




.909 


.913 


.918 


.922 


.927 


.931 


.936 


.940 


.945 


.949 


24 




.954 


.958 


.963 


.968 


.972 


.977 


.982 


.986 


.991 


.996 


25 




.000 


1.005 


1.009 


1.014 


1.019 


1.024 


1.028 


1.033 


1.038 


1.042 


26 




.047 


1.052 


1.057 


1.062 


1.067 


1.071 


1.076 


1.081 


1.086 


1.091 


27 




.096 


1.101 


1.105 


1.110 


1.115 


1.120 


1.125 


1.130 


1.135 


1.140 


28 




.145 


1.150 


1.155 


1.160 


1.165 


1.170 


1.176 


1.181 


1.186 


1.191 


29 




.196 


1.201 


1.206 


1.211 


1.217 


1.222 


1.227 


1.232 


1.238 


1.243 


30 




.248 


1.253 


1.259 


1.264 


1.269 


1.275 


1.279 


1.286 


1.291 


1.296 


31 




.302 


1.307 


1.313 


1.318 


1.324 


1.329 


1.335 


1.340 


1.346 


1.351 


32 




.357 


1.363 


1.368 


1.374 


1.380 


1.386 


1.391 


1.397 


1.403 


1.409 


33 




.415 


1.421 


1.427 


1.432 


1.438 


1.444 


1.450 


1.456 


1.462 


1.469 


34 




.475 


1.481 


1.487 


1.493 


1.499 


1.506 


1.512 


1.518 


1.524 


1.531 


35 




.537 


1.543 


1.549 


1.556 


1.563 


1.569 


1.576 


1.582 


1.589 


1.595 


36 




.602 


1.609 


1.616 


1.622 


1.629 


1.636 


1.643 


1.649 


1.656 


1.663 


37 




.670 


1.677 


1.685 


1.692 


1.699 


1.706 


1.713 


1.720 


1.728 


1.735 


38 




.742 


1.749 


1.757 


1.765 


1.772 


1.780 


1.788 


1.795 


1.803 


1.811 


39 




.819 


1.827 


1.835 


1.843 


1.851 


1.859 


1.867 


1.875 


1.884 


1.892 


40 




.900 


1.909 


1.918 


1.926 


1.935 


1.944 


1.953 


1.962 


1.971 


1.979 


41 




.988 


1.997 


2.007 


2.016 


2.026 


2.035 


2.044 


2.054 


2.064 


2.074 


42 


2 


.083 


2.093 


2.103 


2.114 


2.124 


2.134 


2.145 


2.155 


2.166 


2.177 


43 


2 


.188 


2.199 


2.211 


2.222 


2.234 


2.245 


2.257 


2.269 


2.281 


2.293 


44 


2 


.305 


2.318 


2.331 


2.344 


2.357 


2.370 


2.384 


2.397 


2.411 


2.425 


45 


2.439 


2.453 


2.468 


2.483 


2.498 


2.514 


2.530 


2.546 


2.562 


2.579 


46 


2 


.597 


2.614 


2.631 


2.648 


2.667 


2.686 


2.706 


2.726 


2.746 


2.767 


47 


2 


.789 


2.811 


2.834 


2.857 


2.881 


2.905 


2.932 


2.958 


2.986 


3.015 


48 


3 


.044 


3.077 


3.111 


3.146 


3.182 


3.219 


3.258 


3.300 


3.346 


3.395 


49 
50 


3 


.450 


3.506 


3.571 


3.643 


3.725 


3.820 


3.938 


4.083 


4.275 


4.600 



38 Measurements of Some Achievements in Arithmetic 

TABLE XI 

Per Cent in Each Grade that Solved Each Problem in Addition 

Correctly 

problem grade grade grade grade grade grade grade 



no. 


II 


III 


IV 


V 


VI 


VII 


VIII 


1 


79.4 


74.2 


82.9 


95.2 


98.3 


97.7 


99.5 


2 


88.6 


94.6 


98.9 


99.2 


99.6 


99.6 


99.7 


3 


80.2 


96.4 


98.9 


99.0 


98.9 


99.4 


99.1 


4 


66.7 


76.1 


86.6 


96.0 


97.1 


98.3 


99.3 


5 


66.1 


81.5 


92.1 


98.0 


99.4 


99.7 


100.0 


6 


57.1 


86.2 


93.9 


97.2 


99.2 


99.4 


99.7 


7 


53.0 


87.5 


93.9 


98.8 


99.7 


99.8 


99.7 


8 


45.0 


77.1 


90.1 


96.8 


98.6 


99.4 


100.0 


9 


33.8 


86.2 


94.4 


97.1 


96.9 


96.0 


96.0 


10 


31.1 


86.4 


94.7 


96.5 


98.4 


97.6 


99.1 


11 


38.9 


88.3 


95.9 


96.5 


96.1 


96.6 


98.4 


12 


10.7 


64.9 


89.9 


95.7 


97.9 


97.7 


98.7 


13 


6.6 


37.3 


62.0 


91.3 


98.3 


98.3 


98.7 


14 


7.6 


65.9 


89.9 


96.7 


99.1 


98.2 


98.2 


15 


4.7 


53.3 


82.9 


91.3 


95.1 


95.5 


97.5 


16 


1.2 


38.7 


64.3 


82.6 


84.2 


87.9 


92.0 


17 


4.5 


46.8 


71.6 


80.2 


79.1 


85.8 


87.4 


18 


1.6 


33.8 


64.2 


78.4 


80.8 


87.4 


92.9 


19 


.2 


15.0 


40.9 


58.1 


68.9 


72.2 


84.0 


20 


.2 


14.2 


51.0 


80.8 


92.6 


96.3 


97.1 


21 


.2 


11.6 


45.9 


72.5 


89.1 


91.5 


91.6 


22 




8.0 


33.9 


64.2 


83.4 


88.8 


89.9 


23 




.6 


5.6 


44.9 


77.4 


84.1 


92.0 


24 






.6 


14.4 


46.8 


56.8 


70.7 


25 




.5 


• 2.3 


31.1 


62.7 


71.0 


84.0 


26 




.3 


1.8 


28.0 


66.9 


74.4 


88.8 


27 






1.6 


26.0 


58.3 


74.0 


86.4 


28 






2.3 


24.2 


65.4 


75.6 


82.4 


29 






1.3 


19.1 


47.4 


64.5 


75.2 


30 




.3 


5.5 


22.9 


63.7 


74.6 


85.0 


31 




.5 


5.6 


23.9 


50.1 


53.4 


63.3 


32 






.6 


22.9 


58.9 


73.5 


77.4 


33 






.5 


8.3 


37.1 


50.3 


53.4 


34 




2.4 


21.3 


39.5 


53.4 


74.6 


79.5 


35 




.6 


9.5 


24.6 


50.3 


60.9 


72.1 


36 






.3 


5.8 


28.3 


58.6 


65.1 


37 






.2 


2.9 


27.9 


57.7 


66.0 


38 






.2 


1.3 


24.5 


26.2 


50.4 



to the median point. These values enable us to scale the 
problems. 

By reference to Table XI it is seen that problem No. i was 
solved by 79.4 per cent of the pupils in the second grade. Table 
XII shoves a difiference of 29.4 per cent between the median 
achievement (i.e., a problem solved by 50 per cent of the class) 



Derivation of the Scale 39 

TABLE XII 

Difference Between Fifty Per Cent and the Per Cent in Each Grade 
That Solved Each Problem in Addition Correctly 

PROBLEM grade GRADE GRADE GRADE GRADE GRADE GRADE 



NO. 


II 


III 


IV 


V 


VI 


VII 


VIII 


1 


29.4 


24.2 


32.9 


45.2 


48.3 


47.7 


49.5 


2 


38.6 


44.6 


48.9 


49.2 


49.6 


49.6 


49.7 


3 


30.2 


46.4 


48.9 


49.0 


48.9 


49.4 


49.1 


4 


16.7 


26.1 


36.6 


46.0 


47.1 


48.3 


49.3 


5 


16.1 


31.5 


42.1 


48.0 


49.4 


49.7 


50.0 


6 


7.1 


36.2 


43.9 


47.2 


49.2 


49.4 


49.7 


7 


3.0 


37.5 


43.9 


48.8 


49.7 


49.8 


49.7 


8 


—5.0 


27.1 


40.1 


46.8 


48.6 


49.4 


50.0 


9 


—16.2 


36.2 


44.4 


47.1 


46.9 


46.0 


46.0 


10 


—18.9 


36.4 


44.7 


46.5 


48.4 


47.6 


49.1 


11 


—11.1 


38.3 


45.9 


46.5 


46.1 


46.6 


48.4 


12 


—39.3 


14.9 


39.9 


45.7 


47.9 


47.7 


48.7 


13 


—43.4 


—12.7 


12.0 


41.3 


48.3 


48.3 


48.7 


14 


—42.4 


15.9 


39.9 


46.7 


49.1 


48.2 


48.2 


15 


—45.3 


3.3 


32.9 


41.3 


45.1 


45.5 


47.5 


16 


—48.8 


—11.3 


14.3 


32.6 


38.2 


37.9 


42.0 


17 


—45.5 


—3.2 


21.6 


30.2 


29.1 


35.8 


37.4 


18 


—48.4 


—16.2 


14.2 


28.4 


30.8 


37.4 


42.9 


19 


—49.8 


—35.0 


—9.1 


8.1 


18.9 


22.2 


34.0 


20 


—49.8 


—35.8 


1.0 


30.8 


42.6 


46.3 


47.1 


21 


—49.8 


—38.4 


—4.1 


22.5 


39.1 


41.5 


41.6 


22 




—42.0 


—16.1 


14.2 


33.4 


38.8 


39.9 


23 




—49.4 


—44.4 


—5.1 


27.4 


34.1 


42.0 


24 




—50.0 


—49.4 


—35.6 


—3.2 


6.8 


20.7 


25 




—49.5 


—47.7 


—18.9 


12.7 


21.0 


34.0 


26 




—49.7 


—48.2 


—22.0 


16.9 


24.4 


38.8 


27 




—50.0 


—48.4 


—24.0 


8.3 


24.0 


36.4 


28 




—50.0 


—47.7 


—25.8 


15.4 


25.6 


32.4 


29 




—50.0 


—48.7 


—30.9 


—2.6 


14.5 


25.2 


30 




—49.7 


—44.5 


—27.1 


13.7 


24.6 


35.0 


31 




—49.5 


—44.4 


—26.1 


.1 


3.4 


13.3 


32 




—50.0 


—49.4 


—27.1 


8.9 


23.5 


27.4 


33 




—50.0 


—49.5 


—41.7 


—12.9 


.3 


3.4 


34 




—47.6 


—28.7 


—10.5 


3.4 


24.6 


29.5 


35 




—49.4 


—40.5 


—25.4 


.3 


10.9 


22.1 


36 




—50.0 


—49.7 


—44.2 


—21.7 


8.6 


15.1 


37 




—50.0 


—49.8 


—47.1 


—22.1 


7.7 


16.0 


38 




—50.0 


—49.8 


—48.7 


—25.5 


—23.8 


.4 



and the per cent that actually solved this problem. Table XIII 
shoves that this problem w^ould be located at -1.217 P.E. from 
the median point in the second grade distribution. Table XIII 
shows that this same problem would be located at -.963 P.E. 
from the median of the third grade distribution, etc. Table 



4© Measurements of Some Achievements in Arithmetic 

TABLE XIII 

P.E. Equivalent of Difference Between Fifty Per Cent and the Per 

Cent in Each Grade that Solved Each Problem in 

Addition Correctly 



PROBLEM GRADE 


grade 


GRADE 


GRADE 


GRADE 


GRADE 


GRADE 


NO. 


II 


III 


IV 


V 


VI 


VII 


VIII 


1 


—1.217 


— .963 


—1.409 


—2.468 


—3.146 


—2.958 


—3.820 


2 


—1.788 


—2.384 


—3.395 


—3.571 


—3.938 


—3.938 


—4.083 


3 


—1.259 


—2.667 


—3.395 


—3.450 


—3.395 


—3 . 725 


—3.506 


4 


— .640 


—1.052 


—1.643 


—2.597 


—2.811 


—3.146 


—3.643 


5 


— .616 


—1.329 


—2.093 


—3.044 


—3.725 


—4.083 


—4.600 


6 


— .265 


—1.616 


—2.293 


—2.834 


—3.571 


—3.725 


—4.083 


7 


— .112 


—1.706 


—2.293 


—3.346 


—4.083 


—4.275 


—4.083 


8 


.187 


—1.101 


—1.909 


—2 . 746 


—3.258 


—3 . 725 


—4.600 


9 


.620 


—1.616 


—2.357 


—2.811 


—2.767 


—2.597 


—2.597 


10 


.731 


—1.629 


—2.397 


—2.686 


—3 . 182 


—2.932 


—3.506 


11 


.418 


—1.765 


—2.579 


—2.686 


—2.614 


—2.706 


—3.182 


12 


1.843 


— .567 


—1.892 


—2.546 


—3.015 


—2.958 


—3.300 


13 


2.234 


.480 


— .453 


—2.016 


—3.146 


—3.146 


—3.300 


14 


2.124 


— .608 


—1.892 


—2.726 


—3.506 


—3.111 


—3.111 


15 


2.483 


— .123 


—1.409 


—2.016 


—2.453 


—2.514 


—2.905 


16 


3.346 


.426 


— .543 


—1.391 


—1.757 


—1 . 735 


—2.083 


17 


2.514 


.119 


— .847 


—1.259 


—1.201 


—1.589 


—1.699 


18 


3.182 


.620 


— .539 


—1.165 


—1.291 


—1.669 


—2.177 


19 


4.275 


1.537 


.341 


— .303 


— .731 


— .873 


—1.475 


20 


4.275 


1.589 


— .037 


—1.291 


—2 . 145 


—2.648 


—2.811 


21 


4.275 


1.772 


.153 


— .886 


—1.827 


—2.035 


—2.044 


22 




2.083 


.616 


— .539 


—1.438 


—1.803 


—1.892 


23 




3.725 


2.357 


.190 


—1.115 


—1.481 


—2.083 


24 






3.725 


1.576 


.119 


— .254 


— .807 


25 




3.820 


2.958 


.731 


— .480 


— .820 


—1.475 


26 




4.083 


3.111 


.864 


— .648 


— .972 


—1.803 


27 






3.182 


.954 


— .311 


— .954 


—1.629 


28 






2.958 


1.038 


— .588 


—1.028 


—1.380 


29 






3.300 


1.296 


.097 


— .551 


—1.009 


30 




4.083 


2.370 


1.101 


— .519 


— .982 


—1.537 


31 




3.820 


2.357 


1.052 


.004 


— .164 


— .504 


32 






3.725 


1.101 


— .334 


— .931 


—1.115 


33 






3.820 


2.054 


.489 


— .011 


— .127 


34 




2.932 


1.181 


.395 


— .127 


— .982 


—1.222 


35 




- 3.725 


1.944 


1.019 


— .011 


— .410 


— .869 


36 






4.083 


2.331 


.851 


— .322 


— .575 


37 






4.275 


2.811 


.869 


— .288 


— .612 


38 






4.275 


3.300 


1.024 


.945 


— .015 



XIII thus gives the location of every problem with reference to 
the median point of each grade distribution. The difficulty of 
any problem for any grade can be found by reference to this 
table. 



Derivation of the Scale 41 

4. Measuring the Distance Between the Grades 

Thus far we have located each problem at the proper distance 
from the median point on the base-line of each grade distribution. 
We can now locate the difficulty of each problem for each par- 
ticular grade. We also wish to know how difficult the problems 
are in general. We wish to know what will be the average 
position of each problem when placed on one linear scale. Before 
this can be done, we must determine the distances between the 
consecutive grade medians and we must establish a common 
zero point. 

Three different methods have been used in this study to 
determine the interval between the grade medians. After the 
determinations derived from the three methods were satisfac- 
torily weighted, the average was taken and used as the measure 
of the intergrade interval. For convenience these three methods 
will be called the " problem method," the " quartile method," 
and the " distribution method." 

By the " problem method " the distance between the median 
of two consecutive grades is determined by the difference in 
position each problem holds with reference to the medians of 
two consecutive grade distributions. For example, Table XIII 
shows that problem No. 2 is situated 1.788 P.E. below the 
median of the second grade and 2.384 P.E. below the median 
of the third grade. This makes a difference of .596 P.E. between 
the medians of the second and the third grades so far as this 
problem is concerned. Each problem will give a similar meas- 
ure for the interval between any two consecutive grades. Table 
XIV gives the P.E. intervals between the consecutive grades 
as determined from each addition problem. 

It is interesting to note that as the problems increase in diffi- 
culty larger intervals tend to exist between the grade medians. 
This fact is most clearly brought out by Table XV. In this 
table the determinations of the intergrade intervals from the 
various problems are divided into various groups. The group 
of determinations which is marked below -1.5 P.E. is the aver- 
age of those determinations from Table XIV which came from 
values lower than -1.5 P.E. in Table XIII; the group marked 
-1.5 P.E. to +1.5 P.E. is the average of those determinations 
obtained from values between -1.5 P.E. and +1.5 P.E. ; the 



42 Measurements of Some Achievements in Arithmetic 

TABLE XIV 

P.E. Intervals Shown Between Consecutive Grades by Each 
Addition Problem 

PROBLEM interval INTERVAL INTERVAL INTERVAL INTERVAL INTERVAL 
NO. II-III III-IV IV-V V-VI VI-VII VII-VIII 



1 


— .254 


.446 


1.059 


.678 


— .188 


.862 


2 


.596 


1.011 


.176 


.367 


.000 


.145 


3 


1.408 


.728 


.055 


— .055 


.330 


— .219 


4 


.412 


.591 


.944 


.214 


.335 


.497 


5 


.713 


.764 


.951 


.681 


.358 


.517 


6 


1.351 


.677 


.541 


.737 


.154 


.358 


7 


1.594 


.587 


1.053 


.737 


.192 


— .192 


8 


1.288 


.808 


.837 


.512 


.467 


.875 


9 


2.236 


.741 


.454 


— .044 


— .170 


.000 


10 


2.360 • 


.768 


.289 


.496 


— .250 


.574 


11 


2.183 


.814 


.107 


— .072 


.092 


.476 


12 


2.410 


1.325 


.654 


.469 


— .057 


.342 


13 


1.754 


.933 


1.563 


1.130 


.000 


.154 


14 


2.732 


1.284 


.834 


.680 


— .395 


.000 


15 


2.606 


1.286 


.607 


.437 


.061- 


.391 


16 


2.920 


.969 


.848 


.366 


— .022 


.348 


17 


2.395 


.966 


.412 


— .058 


.388 


.110 


18 


2.562 


1.159 


.626 


.126 


.378 


.508 


19 


2.738 


1.196 


.644 


.428 


.142 


.602 


20 


2.686 


1.626 


1.254 


.854 


.503 


.163 


21 


2.503 


1.619 


1.039 


.941 


.206 


.009 


22 




1.467 


1.155 


.899 


.365 


.089 


23 




1.368 


2.167 


1.305 


.366 


.602 


24 






2.149 


1.457 


.373 


.553 


25 




.862 


2.227 


1.211 


.340 


.655 


26 




.972 


2.247 


1.512 


.324 


.831 


27 






■ 2.228 


1.265 


.643 


.675 


28 






1.920 


1.896 


.440 


.352 


29 






2.004 


1.199 


.648 


.458 


30 




1.730 


1.269 


1.620 


.463 


.555 


31 




1.463 


1.305 


1.048 


.168 


.340 


32 






2.624 


1.435 


.597 


.184 


33 






1.766 


1.565 


.500 


.116 


34 




1.751 


.788 


.522 


.855 


.240 


35 




1.781 


.925 


1.030 


.399 


.459 


36 






1.752 


1.480 


1.173 


.253 


37 






1.464 


1.942 


1.157 


.324 


38 






.975 


2.276 


.079 


.960 



group marked above +1.5 P.E. is the average of the determina- 
tions obtained from values larger than H-i.5 P.E. The group 
marked " select group " is the average of the determinations 
obtained from values between -2 P.E. and +2 P.E. The group 
marked " compositive " average is the average of the determina- 
tions in Table XIV. 



Derivation of the Scale 43 

TABLE XV 

Averages of Groups of Determinations of Intergrade Intervals 
AS Measured by the Addition Problems 

II-III III-IV iv-v V-VI VI-VII VII-VIII 

GROUP inter- inter- INTER- INTER -INTER- INTER- 

VAL VAL VAL VAL VAL VAL 

Below 1.5 P.E 1.695 .842 .675 .508 .131 .347 

—1.5 P.E. to 

-f 1.5 P.E 539 .959 .846 1.028 .509 .423 

Above 1.5 P.E 2.531 1.456 1.801 1.744 1.522i .6OO1 

Compositve Average. . 1.866 1.099 1.155 .876 .300 .374 

Select Group 1.429 1.093 .854 1.011 .456 .437 

^ These two values have been estimated. 

Table XV show^s so far as this list of problems is concerned 
that the greatest difference between the medians of the different 
grades is brought about by the more difficult problems and that 
the least amount of difference is brought about by the least 
difficult problems. The one exception to this general statement 
is found in group --1.5 P.E. to +1.5 P.E. for the interval be- 
tween the second and third year. The smallness of this deter- 
mination is due to the small number of cases that happened to 
fall within those middle limits. These same facts are brought 
out in similar tables for the other fundamental processes. These 
results are in conformity with the results found by Dr. Trabue 
in his measurement with completion-test language scales. 

Keeping in mind the fact that the greatest difference between 
the medians of any two consecutive grades is brought about by 
the most difficult problems and also that the least difference is 
brought about by the easier problems, it would seem that the 
best measure of the interval would be the average of those 
determinations that come from near the median. It seems rather 
unfair that those problems at the extreme ends of a distribution 
should have equal weight with those near the middle of the 
distribution. In order to give more weight to the problems near 
the median of the distribution the average of those determina- 
tions in Table XIV which were obtained from values -2 P.E. 
to -f2 P.E. in Table XIII will be used in addition to the com- 
posite average in computing the final measure of the intervals 
between the grade medians. This last group of determinations 
is designated as the " select group " in Table XV. Thus the 
composite average and the average of the determination of the 
" select group " are both measures of the intergrade intervals 



44 Measurements of Some Achievements in Arithmetic 

and both are derived from the " problem method " and will be 
used in the final determination of the intergrade intervals. 

The second method of determining the distance between the 
grades was previously designated as the " quartile " method. 
If we have a normal surface of distribution, as we have assumed, 
the quartile of any distribution should be equal to the P.E. of 
that distribution. Therefore, if we divide the quartile of a 
distribution into the crude score intervals, we will get the inter- 
val between the medians of the grades in terms of P.E. Since 
for each interval between the grades there are two quartile 
measures, the average of the two quartiles is used as a divisor 
of the crude score interval between the grades. 

Table XVI shows the intervals obtained by this process. 



TABLE XVI 
Determination of Quartile Intervals Between Grades 





grade 
II 


grade 

III 


grade 

IV 


GRADE 
V 


GRADE 
VI 


GRADE 
VII 


GRADE 
VIII 


Median . 
Quartile. 


6.819 
2.712 


14.509 
2.996 


18.321 
2.247 


23.073 

2.837 


29.774 
3.910 


32.446 
3.099 


33.987 
2.118 



Score 

Interval . 
Average 
tf: Interval . 
Quartile 

Interval . 



7.690 


3.812 


4.752 


6.701 


2.672 


1.541 


2.854 


2.621 


2.542 


3.374 


3.505 


2.609 


2.694 


1.459 


1.869 


1.986 


.762 


.591 



This table is made from data taken from Table VIII. The score 
interval or crude score between the second and third grade 
medians is the difference between the median number of prob- 
lems solved in the second and third grades. The average quar- 
tile is the average of the quartiles of the second and third grade 
distributions. The crude score (7.690) divided by the average 
quartile (2.854) gives the quartile interval (2.694), the distance 
between the second and third grade medians. By hypothesis 
this quartile interval is in terms of P.E. 

Third measure of the interval between the median of the 
different grades was by means of the " distribution method." 
It is based upon the amount of overlapping of the consecutive 
grade distributions. Table VIII shows that there are pupils in the 



Derivation of the Scale 45 

second grade that excel the median achievement of the pupils 
in the third grade. On the other hand, there are pupils in the 
third grade that do not reach the median achievement of the 
second grade pupils. Between the median of the second grade 
distribution (6.819) and the median of the third grade (14.509) 
distribution lie 46.22 per cent of the 489 cases in the second 
grade distribution; between the same medians lie 42.16 per 
cent of the 615 cases in the third grade distributions. Since 
these percentages are deviations from the median or 50 per cent 
they can be turned into P.E. values by referring to Table X. 
Thus, as determined from the second grade distribution, the 
interval between the second and third grade medians is 2.643 
P.E. As determined by the third grade distribution the interval 
between the third and second grade medians is 2.099 P-E. Thus 
we have two direct measures for this same interval. Similarly, 
direct measures can be made for the intervals between each 
of the other succeeding grades. 

By the same reasoning, between the second grade median and 
the fourth grade median lie 49.79 per cent of the second grade 
distribution; between the third grade median and the fourth 
grade median lie 35.98 per cent of the third grade distribution. 
Turning the percentages into P.E. we find that the fourth 
grade median is 4.256 P.E. above the second grade median and 
1.601 P.E. above the third grade median. If we take the dis- 
tance between the third and fourth grade median from the dis- 
tance between the second and fourth grade medians, we get an 
indirect measure of the distance between the second and third 
grade medians. Thus 4.256 P.E. -1.601 P.E. = 2.655 P.E., the 
indirect measure of the interval between the second and third 
grades. 

Table XVII shows the percentage with P.E. equivalent of each 
grade lying between the median and the medians of the neigh- 
boring grades. 

Various other indirect measures can be obtained in a similar 
way. Indeed, if one wished he could get a still further remote 
indirect measure of this same second and third grade interval 
by taking the distance between the third and fifth grade median 
from the distance between the second and fifth grade medians. 
This latter determination would, in the writer's opinion, be influ- 



4-6 Measurements of Some Achievements in Arithmetic 

enced so much by the extreme ends of the distributions that it 
should be given no weight in the final determination of the 
grade intervals. 

In the final determination of the grade intervals by the " dis- 
tribution method " the direct measures were felt to be the best 
measures, and hence they were given double weight while the 
indirect measures were given but single weight. 



TABLE XVII 

Percentage with P.E. Equivalent of Each Grade Lying Between the 
Median and the Medians of the Neighboring Grades 





II 


HI 


IV 


v 


VI 


VII 


VIII 


11% 

P.E. 


46.22 
2.634 


49.79 
4.256 












111% 
P.E. 


42.16 
2.099 




35.98 
1.601 


49.38 
3.709 








iv% 

P.E. 


48.89 
3.390 


36.34 
1.625 




41.26 
2.012 


49.83 
4.373 






v% 

P.E. 




49.34 
3.676 


40.57 
1.950 




43.17 
2.207 


48.33 
3.157 




vi% 

P.E. 






48.96 
3.428 


38.44 
1.775 




18.43 
.711 


29.35 
1.214 


VII % 
P.E. 








47.16 

2.825 


19.45 

.754 




14.32 
.544 


VIII % 
P.E. 










36.30 
1.622 


18.47 
.713 





TABLE XVIII 

Determinations of the Intergrade Intervals from Over- 
lapping OF Distributions in Addition 



determination 


ii-iii 


III-IV 


IV-V 


V-VI 


VI-VII 


VII-VIII 


Lower Indirect. . . . 

Lower Direct 

Lower Direct 

Upper Direct 

Upper Direct 

Upper Indirect .... 

Total 


2.099 
2.099 
2.634 
2.634 
2.655 

12.121 


1.291 
1.625 
1.625 
1.601 
1.601 
1.697 

9.440 


2.051 
1.950 
1.950 
2.012 
2.012 
2.166 

12.141 


1.478 
1.775 
1.775 
2.207 
2.207 
2.446 

11.888 


1.050 

.754 
.754 
.711 
.711 
.670 

4.650 


.868 
.713 
.713 
.544 
.544 

3.382 


Average 


2.424 


1.573 


2.023 


1.981 


.775 


.676 







Derivation of the Scale 



47 



Table XVIII shows the interval between the grades as deter- 
mined by the distribution method. 

Thus we have the results of the intergrade intervals as deter- 
mined by the three different methods. All of the determinations 
have about the same general characteristics. The writer felt 
that the " select group " of determination, which is based upon 
the individual problems not varying over 2 P.E. in difficulty 
from the median problem of the grade, was the best measure. 
He also felt that the second best measure was the composite 
average based upon the difficulty of all the individual problems. 
Both the " selected group " determinations and the composite 
averages take into consideration the fact that the problems are 
not of the same difficulty. Hence it appeared to the writer that 
it might be fair to give these determinations increased weight 
in making the final determination of the intergrade intervals. 
Thus the " selected group " determinations were given a weight 
of three, and the " composite average " determinations a weight 
of two while the determinations from the quartile method and 
those from the " distribution method " were given but a single 

TABLE XIX 
Final Determination of Intervals Between Successive Grades 



METHOD 


ii-iii 


III-IV 


IV-V 


V-VI 


VI-VII 


VII-VIII 


Prob. select group . 


1.429 


1.093 


.854 


1.011 


.456 


.437 


a u u 


1.429 


1.093 


.854 


1.011 


.456 


.437 


u tl u 


1.429 


1.093 


.854 


1.011 


.456 


.437 


" composite av. 


1.866 


1.099 


1 . 155 


.876 


.300 


.374 


M U U 


1.866 


1.099 


1.155 


.876 


.300 


.374 


Distribution 


2.424 


1.573 


2.023 


1.981 


.775 


.676 


Quartile 


2.694 


1.459 


1.869 


1.986 


.762 


.591 


Average 


1.877 


1.216 


1.252 


1.250 


.501 


.475 



weight. Table XIX gives the average of the determinations 

which is used as the final measure of the intergrade intervals in 
the construction of the Addition Scales.'- 

1 The other intergrade intervals as used in the final development of the 
other scales are as follows : 

II-III III-IV IV-V V-VI VI-VII VII-VIII 

Subtraction 1.646 1.073 1.392 1.233 .751 .820 

Multiplication 1.748 1.636 1.491 .822 .536 

Division 1.014 1.554 1.368 .667 .659 



48 Measurements of Some Achievements in Arithmetic 

5. Location of the Zero Point 

Having determined the distance between the various grades, it 
is easy to locate all of the problems in terms of any grade median. 
However, if we wish to know the exact relation of one problem 
to another, if we wish to know how many times more difficult 
one problem is than another, it is necessary to find the location 
of each problem with reference to a common zero point. 

Professor Thorndike in his " Mental and Social Measure- 
ments " (p. 16) in speaking of the definition of a zero point 
says, " The zero point may be absolute, measuring just not any 
of the thing, or arbitrary, meaning a point called zero though 
actually designating some amount of the thing. Thus the thing 
being temperature, 20° C is 20° above the arbitrary zero — the 
melting of ice — and 293° above the supposed absolute zero, just 
not any molecular motion in a gas." 

The zero point in connection with any one of "these scales is 
an arbitrary one. It means simply " the inability to solve cor- 
rectly a single problem " as presented under the standard con- 
ditions for giving these tests. It does not mean that a child has 
absolutely no ability in addition or subtraction or multiplication 
or division. It is probable that if the problems had been pre- 
sented orally to the pupils in the lower grades they would have 
solved more problems correctly. Some of the pupils who showed 
" inability to solve correctly a single problem " as the test was 
presented to them, no doubt would have solved a few problems 
if presented orally. Moreover, it should be added that zero 
ability in division (i.e., inability to solve a single problem) does 
not mean that a child will have zero ability in addition. It 
should also be added that we cannot say, as the scales are in 
their present condition, that a problem with a value of i in one 
process is equal in difficulty to a problem with the same value in 
another process. Each scale has its own zero point. We there- 
fore cannot treat values as equal which have been developed 
from different zero points. 

The method for locating the zero point is the same for all 
the fundamental processes, but I shall deal in detail only with 
addition. Table VIII shows that of the 489 pupils in the second 
grade 44.88 per cent lie between those children who could not 
get a single problem and the median achievement for that grade 



Derivation of the Scale 49 

(6.819). Since this represents deviation from the median or 50 
percentile, reference to Table X shows that a deviation of 44.88 
per cent from the median of a normal distribution represents 
a distance of 2.322 P.E. This means then that the median of 
the second grade in addition is 2.422 P.E. above no score at all. 

Table VIII also shows that of 615 pupils in the third grade 49.34 
per cent lie between those children who could not get a single 
problem and the median achievement for that grade (14.509). 
This would locate the median for the third grade 3.676 P.E. 
above the zero point. We have already determined that the 
median of the third grade is 1.877 P-E. (Table XIX) above 
that of the second grade. By subtracting the distance the third 
grade median is above the second from the distance the third 
grade median is above the zero point, we get a measure of the 
distance the second grade median is above the zero point. Thus 
3.676 P.E. minus 1.877 P-E. = 1-799 P-E., another measure of 
the distance that the second median is above the zero point. 

Table VIII also shows that the median achievement of the 
second grade distribution is 6.819 problems and that the quartile 
is 2.712 problems. Since we have assumed a normal surface 
of frequency, the quartile is equal to the P.E. ; thus by dividing 
the median achievement by the quartile of the second grade we 
find that the median of the second grade is 2.514 P.E. above 
zero. 

Similarly the median achievement for the third grade is 14.509 
problems and the quartile 2.996 problems. By dividing the 
median achievement by the quartile we find that the third grade 
median is 4.843 P.E. above zero. Subtracting the distance the 
third grade median is above the second (1.877 P.E.) from the 
distance the third grade is above zero (4.843 P.E.) gives us the 
distance the second grade median is above zero (2.966 P.E.) 

Thus we have four determinations of our zero point as follows : 

From the second grade distribution 2 .422 

" third " " 1.799 

" " second " achievement 2 .514 

" " third " « 2.966 

Average 2.425 

The average of the four determinations probably represents 
a better measure of the distance than any single measure, so for 



50 Measurements of Some Achievem^ents in Arithmetic 

the addition scale we shall use the point 2.425 P.E. below the 
second grade median as the arbitrary zero point. 

Since we have determined the distance between the medians 
of the various grades (Table XIX) and know that the second 
grade median is 2.425 P.E. above zero it is easy to determine 
the distance each grade median is above zero. 

TABLE XX 
Distance the Median of Each Grade in Addition is Above Zero 

1 = P.E. 



GRADE 


ABOVE ZERO 


BELOW NEXT GRADE 


II 


2.425 


1.877 


III 


4.302 


1.216 


IV 


5.518 


1.252 


V 


6.770 


1.250 


VI 


8.020 


.501 


VII 


8.521 


.475 


VIII 


8.996 


" 



Figure 11 (page 51) represents graphically the relations of 
the grade distribution to each other, the relations of the grade 
medians to each other and to the zero point, as determined by 
the values of Table XX and based upon the assumption that 
achievement in the solution of problems is distributed normally. 

6. Referring All the Problems in Addition to Zero 

Table XIII gives the value of each problem in addition for 
each grade. It shows, for instance, that problem No. i has a 
negative value of 1.217 P.E. in the second grade and a negative 
value of .963 P.E. in the third grade, etc. By reference to Table 
XX one finds that the second grade median is 2.425 P.E. and 
the third grade median is 4.302 P.E. above zero. By subtract- 
ing 1. 217 from 2.425 and .903 from 4.302 we find that problem 
No. I, in the second and third grades, is respectively 1.208 and 
3.339 P.E. above zero. Wherever the value in Table XIII is 
positive instead of negative, as these just cited have been, add 
this value to the value which that particular grade median is 
above zero. For example, to determine the distance problem 
No. 8 is above zero in the second grade we add .187 P.E. to 
2.425 P.E. Thus we find that problem No. 8 is 2.612 P.E. above 
zero in the second grade. Table XXI shows the location above 
zero of each problem in addition. 



Derivation of the Scale 



51 



w 



o 



oi 



52 Measurements of Some Achievements in Arithmetic 

Since we have determined the location above zero of each 
problem in each grade we are now ready to determine the gen- 
eral value of each problem and to locate it on a linear scale. 
In order to do this we must find the position which best repre- 
sents the difficulty of each problem. 

TABLE XXI 
Location Above Zero of Each Addition Problem 

problem grade grade grade grade grade grade grade 



NO. 


II 


III 


IV 


V 


VI 


VII 


VIII 


1 


1.208 


3.339 


4.109 


4.302 


4.874 


5.563 


5.176 


2 


.637 


1.918 


2.123 


3.199 


4.082 


4.583 


4.913 


3 


1.166 


1.635 


2.123 


3.302 


4.625 


4.796 


5.490 


4 


1.785 


3.250 


3.875 


4.173 


5.209 


5.375 


5.353 


5 


1.809 


2.973 


3.425 


3.726 


4.295 


4.438 


4.396 


6 


2.160 


2.686 


3.225 


3.936 


4.449 


4.796 


4.913 


7 


2.313 


2.596 


3.225 


3.424 


3.937 


4.246 


4.913 


8 


2.612 


3.201 


3.609 


4.024 


4.762 


4:796 


4.396 


9 


3.045 


2.686 


3.161 


3.959 


5.253 


5.924 


6.399 


10 


3.156 


2.673 


3.121 


4.084 


4.838 


5.589 


5.490 


11 


2.843 


2.537 


2.939 


4.084 


5.406 


5.815 


5.814 


12 


4.268 


3.735 


3.626 


4.224 


5.005 


5.563 


5.696 


13 


4.659 


4.782 


5.065 


4.754 


4.874 


5.375 


5.696 


14 


4.549 


3.694 


3.626 


4.044 


4.514 


5.410 


5.885 


15 


4.908 


4.179 


4.109 


4.754 


5.567 


6.007 


6.091 


16 


5.771 


4.728 


4.975 


5.379 


6.263 


6.786 


6.913 


17 


4.939 


4.421 


4.671 


5.511 


6.819 


6.932 


7.297 


18 


5.607 


4.922 


4.979 


5.605 


6.729 


6.852 


6.819 


19 


6.700 


5.839 


■ 5.859 


6.467 


7.289 


7.648 


7.521 


20 


6.700 


5.891 


5.481 


5.479 


5.875 


5.873 


6.185 


21 


6.700 


6.074 


5.671 


5.884 


6.193 


6.486 


6.952 


22 




6.385 


6.134 


6.231 


6.582 


6.718 


7.104 


23 




8.027 


7.875 


6.960 


6.905 


7.040 


6.913 


24 






9.243 


8.346 


8.139 


8.267 


8.189 


25 




8.122 


8.476 


7.501 


7.540 


7.701 


7.521 


26 




8.385 


8.629 


7.634 


7.372 


7.549 


7.193 


27 






8.700 


7.724 


7.709 


7.567 


7.367 


28 






8.476 


7.808 


7.432 


7.493 


7.616 


29 






8.818 


8.066 


8.117 


7.970 


7.987 


30 




8.385 


7.888 


7.871 


7.501 


7.539 


7.459 


31 




8.122 


7.875 


7.822 


8.024 


8.357 


8.492 


32 






9.243 


7.871 


7.686 


7.590 


7.881 


33 






9.338 


8.824 


8.509 


8.510 


8.869 


34 




7.234 


6.699 


7.165 


7.893 


7.539 


7.774 


35 




8.027 


7.462 


7.789 


8.009 


8.111 


8.127 


36 






9.601 


9.101 


8.871 


8.199 


8.421 


37 






9.793 


9.581 


8.889 


8 233 


8.384 


38 






9.793 


10.070 


9.044 


9.166 


8.981 



Derivation of the Scale 53 

In making this determination it was felt that the truest value 
of any problem came from the distribution where the median 
achievement was nearest the location of that problem. It was 
also felt that those values which came from those distributions 
where the median achievements were farthest from the location 
of that problem should have little or no weight in the deter- 
mination of the final value of the problem. 




-3 P.E. -I P.E. M +1 P.E. +3 P.E. 

Fig. 12. Showing the influence of various parts of a distribution in 
determining the general value of a problem. 

Thus, as represented by Fig. 12, in computing the general 
final value of a problem in these various distributions, that value 
from each grade distribution is given double weight if the prob- 
lem is less than i P.E. distance from the median achievement 
of that distribution; single weight, if it is more than i P.E. 
but less than 3 P.E. distance from the median achievement of 
the distribution ; and not considered at all if the problem is more 
than 3 P.E. distance from the median achievement of the dis- 
tribution. The average of all of these determinations is taken 
as the general difficulty of a problem. 

Therefore, in making this final determination of the general 
value of any problem reference must be made to both Table 
XIII and Table XXI. Table XIII is used to determine the 
weight that shall be given to the value in Table XXI. For 
example, Table XIII shows that problem No. i is located at 
.963 P.E. from the median achievement of the third grade. It 
also shows that this same problem is located at more than i P.E. 
but less than 3 P.E. distance from the median achievement in 
grades two, four, and five. With these facts in mind weight 
the value given in Table XXI accordingly. The value in the 
third grade would have double weight, while the values in the 



54 Measurements of Some Achievements in Arithmetic 

second, fourth, and fifth grades would have single weight. The 
other values are disregarded. The average of the above deter- 
minations represents the difficulty of the problem. 

Table XXII gives the general value, i.e., distance above the 
arbitrary zero, for each problem in addition. These values are 
final and are used in locating the problems on the linear scale. 

TABLE XXII 
Final Value of Addition Problems 



ANK 


no. of problem 


VALUE 


1 


2 


1.23 


2 


3 


1.40 


3 


5 


2.50 


4 


7 


2.61 


5 


6 


2.83 


6 


8 


3.21 


7 


1 


3.26 


8 


4 


3.35 


9 


10 


3.63 


10 


11 


3.78 


11 


14 


3.92 


12 


9 


4.18 


13 


12 


4.19 


14 


13 


4.85 


IS 


15 


4.97 


16 


17 


5.52 


17 


16 


5.59 


18 


18 


5.73 


19 


20 


5.75 


20 


21 


6.10 


21 


22 


6.44 


22 


19 


6.79 


23 


23 


7.11 


24 


34 


7.43 


25 


26 


7.47 


26 


30 


7.61 


27 


27 


7.62 


28 


25 


7.67 


29 


28 


7.71 


30 


32 


7.71 


31 


35 


7.97 


32 


29 


8.04 


33 


31 


8.18 


34 


24 


8.22 


35 


36 


8.58 


36 


37 


8.67 


37 


33 


8.67 


38 


38 


9.19 



Note. — These values are listed in Part I of this monograph. The graphic representa- 
tion of the scale, i. e., the problem placed on a linear projection according to these final 
values, is also given in Part I. 



Section II. TABLES OF CRUDE DATA FROM WHICH 
SCALES WERE DEVELOPED 









TABLE XXIII 








Distribution According to the Number of Subtraction 










Problems Solved 










grade i 


GRADE 1 


GRADE i 


GRADE 


GRADE 


GRADE ( 


3RADE 




II 


III 




IV 


V 


VI 


VII 


VIII 


35 












6 


14 


34 


34 












8 


42 


45 


33 












8 


51 


56 


32 












23 


76 


68 


31 










2 


27 


68 


80 


30 










1 


25 


79 


75 


29 













41 


86 


57 


28 










2 


39 


90 


48 


27 










13 


38 


66 


33 


26 










13 


46 


76 


15 


25 










19 


68 


70 


18 


24 










33 


70 


34 


7 


23 








5 


57 


63 


46 


9 


22 




2 




14 


61 


59 


52 





21 




1 




13 


84 


36 


23 


5 


20 




7 




42 


101 


36 


27 


2 


19 




24 




51 


72 


17 


6 


2 


18 


1 


26 




54 


68 


24 


5 


1 


17 





24 




54 


51 


14 


4 




16 


1 


17 




51 


36 


7 


3 




15 


1 


28 




58 


17 










14 


2 


36 




49 


13 


2 







13 


8 


52 




60 


7 


1 







12 


7 


49 




38 


10 


1 







11 


11 


54 




27 


8 


1 


1 




10 


15 


53 




20 


3 


1 






9 


23 


58 




14 


2 









8 


28 


29 




6 


1 









7 


33 


26 




11 


1 









6 


31 


21 




3 


2 









5 


34 


13 




5 












4 


19 


21 




2 





1 






3 


13 


10 




1 


2 








2 


20 


11 




4 


1 








1 


30 


21 




6 


1 











104 


33 




18 


3 








No. Tested. 


381 


616 




606 


684 


662 


919 


555 


Median 


5.132 


11. 


223 


15.672 


20.436 


24.971 


28.517 


31.687 


25 Per Cent. 


.916 


6. 


923 


12.908 


18.191 


22.415 


25.411 


28.974 


75 Per Cent. 


8.063 


14. 


306 


18.509 


22.492 


28.295 


31.313 


32.945 


Quartile . . . 


3.574 


3. 


692 


2.801 


2.651 


2.940 


2.951 


1.986 
55 



56 Measurements of Some Achievements in Arithmetic 

TABLE XXIV 

Number in Each Grade that Solved Each Problem in 
Subtraction Correctly 

problem grade grade grade grade grade grade grade 

no. II III IV V VI VII VIII 



1 


225 


464 


556 


664 


658 


917 


555 


2 


133 


326 


449 


625 


646 


906 


553 


3 


222 


529 


573 


675 


662 


919 


555 


4 


199 


496 


568 


665 


654 


904 


555 


5 


169 


469 


551 


668 


660 


917 


555 


6 


184 


511 


534 


671 


653 


917 


555 


7 


158 


468 


547 


658 


651 


913 


552 


8 


80 


358 


445 


614 


634 


908 


551 


9 


113 


425 


510 


646 


650 


909 


553 


10 


72 


391 


496 


627 


639 


894 


545 


11 


52 


339 


466 


635 


645 


907 


553 


12 


53 


363 


498 


649 


651 


910 


552 


13 


54 


364 


509 


655 


655 


910 


553 


14 


13 


222 


362 


592 


625 


885 


546 


15 


9 


193 


358 


588 


638 


907 


551 


16 


2 


141 


315 


547 


570 


870 


534 


17 


2 


145 


334 


566 


573 


840 


512 


18 





78 


244 


498 


586 


849 


542 


19 


1 


61 


161 


389 


496 


776 


515 


20 




31 


121 


326 


403 


662 


441 


21 




8 


75 


326 


492 


772 


519 


22 




14 


53 


285 


434 


758 


518 


23 







2 


27 


176 


342 


223 


24 







1 


81 


240 


417 


277 


25 




. 





53 


197 


340 


223 


26 




3 


75 


228 


494 


758 


518 


27 






12 


204 


467 


743 


494 


28 









125 


304 


528 


420 


29 









24 


199 


565 


472 


30 






2 


17 


181 


438 


438 


31 








15 


140 


361 


312 


32 








10 


160 


462 


411 


33 








17 


177 


445 


359 


34 








17 


199 


450 


375 


35 








8 


115 


336 


389 



No. Tested 381 616 606 684 662 919 555 



Data From Which Scales Were Developed 57 

TABLE XXV 
Location Above Zero of Each Subtraction Problem 



OBLEM 


: GRADE 


GRADE 


GRADE 


GRADE 


GRADE 


GRADE 


GRADE 


FINAL 


NO. 


II 


III 


IV 


V 


VI 


VII 


VIII 


VALUE 


1 


.782 


1.748 


1.781 


2.416 


2.735 


2.936 


3.431 


1.501 


2 


1.691 


2.650 


2.877 


3.201 


3.528 


3.953 


4.093 


2.645 


3 


.805 


1.167 


1.451 


1.927 


1.860 


2.611 


3.431 


1.057 


4 


1.034 


1.487 


1.566 


2.393 


3.114 


4.029 


3.431 


1.502 


5 


1.325 


1.710 


1.856 


2.269 


2.377 


2.936 


3.431 


1.697 


6 


1.179 


1.347 


2.086 


2 . 150 


3.202 


2.936 


3.431 


1.447 


7 


1.434 


1.355 


1.909 


2.596 


3.314 


3.486 


4.211 


1.745 


8 


2.312 


2.459 


2.908 


3.343 


3.898 


3.865 


4.388 


2.898 


9 


1.906 


2.027 


2.348 


2.857 


3.349 


3.761 


4.093 


2.178 


10 


2.423 


2.250 


2.489 


3.173 


3.774 


4.354 


4.920 


2.959 


11 


2.738 


2.572 


2.744 


3.061 


3.579 


3.911 


4.093 


2.877 


12 


2.725 


2.425 


2.467 


2.802 


3.314 


3.705 


4.211 


2.568 


13 


2.705 


2.421 


2.354 


2.665 


3.010 


3.705 


4.093 


2.513 


14 


3.822 


3.289 


3.471 


3.584 


4.103 


4.563 


4.849 


3.699 


15 


4.074 


3.485 


3.494 


3.625 


3.793 


3.911 


4.388 


3.635 


16 


4.936 


3.863 


3.757 


3.984 


4.851 


4.814 


5.400 


4.346 


17 


4.936 


3.833 


3.645 


3.824 


4.817 


5.185 


5.917 


4.409 


18 




4.454 


4.199 


4.327 


4.680 


5.087 


5.073 


4.418 


19 


5.391 


4.671 


4.762 


4.969 


5.464 


5.712 


5.865 


5.182 


20 




5.201 


5.088 


5.312 


6.050 


6.342 


6.172 


5.763 


21 




6.062 


5.548 


5.312 


5.492 


5.730 


5.786 


5.524 


22 




5.748 


5.851 


5.538 


5.864 


5.825 


5.809 


5.754 


23 






7.918 


7.841 


7.432 


7.695 


8.399 


7.841 


24 






8.435 


6.984 


6.979 


7.383 


8.068 


7.406 


25 








7.341 


7.246 


7.707 


8.399 


7.720 


26 




6.700 


5.548 


5.867 


5.478 


5.825 


5.809 


5.696 


27 






7.341 


6.013 


5.661 


5.915 


6.204 


5.911 


28 








6.567 


6.609 


6.931 


6.998 


6.774 


29 








7.913 


7.233 


6.777 


6.488 


7.070 


30 






7.918 


8.159 


7.355 


7.296 


6.840 


7.383 


31 








8.242 


7.646 


7.614 


7.800 


7.694 


32 








8.485 


7.498 


7.200 


7.073 


7.208 


33 








8.159 


7.382 


7.270 


7.472 


7.486 


34 








8.159 


7.233 


7.248 


7.354 


7.412 


35 








8.622 


7.851 


7.719 


7.249 


7.517 



58 Measurements of Some Achievements in Arithmetic 

TABLE XXVI 

Distribution According to the Number of Multiplication 
Problems Solved 



GRADE 





Ill 


IV 


V 


VI 


VII 


VIII 


39 










7 


9 


38 










9 


19 


37 








5 


25 


41 


36 








10 


34 


36 


35 








8 


45 


51 


34 








14 


72 


56 


33 








27 


63 


68 


32 








24 


97 


41 


31 








36 


84 


56 


30 






1 


41 


86 


44 


29 






3 


45 


64 


45 


28 






7 


41 


66 


29 


27 






10 


43 


63 


24 


26 






10 


45 


52 


22 


25 




1 


18 


42 


38 


7 


24 




2 


19 


42 


29 


4 


23 




1 


34 


45 


27 


4 


22 




4 


32 


44 


19 


1 


21 




3 


40 


29 


18 


3 


20 




8 


55 


27 


10 


2 


19 




16 


73 


14 


7 





18 


1 


14 


62 


16 


10 


1 


17 





24 


79 


14 


5 


1 


16 


2 


25 


44 


16 


3 





15 





38 


37 


8 


6 


1 


14 





38 


44 


11 


1 




13 


7 


49 


27 


5 


1 




12 


12 


41 


20 


3 


2 




11 


17 


42 


21 


3 






10 


20 


46 


15 


1 






9 


17 


45 


5 


1 






8 


27 


37 


8 


2 






7 


27 


29 


4 









6 


28 


36 


3 









5 


44 


36 


8 


2 






4 


91 


29 


2 









3 


52 


19 












2 


34 


4 


1 









1 


22 


4 





1 









55 


13 


7 








No. Tested . . . 


456 


604 


689 


665 


943 


565 


Median 


4.714 


11.095 


18.315 


26.144 


30.587 


32.940 


25 Per Cent , . 


3.058 


7.345 


15.196 


22.301 


27.123 


29.940 


75 Per Cent . . 


7.593 


14.605 


21.044 


29.973 


33.306 


35.289 



Quartile. 



2.267 



3.630 2.924 



3.836 



3.091 2.674 



Data From Which Scales Were Developed 



59 



TABLE XXVII 

Number in Each Grade that Solved Each Problem in 
Multiplication Correctly 



PROBLEM 


grade 


grade 


GRADE 


GRADE 


GRADE 


GRADE 


NO. 


III 


IV 


V 


VI 


VII 


VIII 


1 


372 


577 


673 


660 


936 


563 


2 


373 


570 


671 


657 


938 


561 


3 


338 


545 


676 


658 


936 


565 


4 


281 


529 


661 


652 


936 


563 


5 


157 


445 


644 


656 


935 


559 


6 


137 


421 


644 


645 


937 


562 


7 


139 


462 


638 


648 


932 


562 


8 


134 


447 


651 


649 


933 


562 


9 


90 


349 


602 


613 


898 


547 


10 


82 


357 


593 


615 


899 


547 


11 


34 


284 


571 


603 


893 


548 


12 


40 


273 


553 


589 


868 


527 


13 


18 


198 


453 


526 


820 


514 


14 


6 


173 


433 


485 


768 


506 


IS 


5 


149 


439 


571 


880 


545 


16 


8 


192 


495 


555 


823 


491 


17 





14 


228 


430 


844 


541 


18 


3 


100 


408 


503 


770 


501 


19 


2 


85 


400 


531 


801 


509 


20 


3 


68 


251 


330 


593 


402 


21 





9 


172 


398 


820 


528 


22 





5 


123 


418 


821 


542 


23 


4 


56 


248 


515 


834 


547 


24 





36 


195 


476 


834 


539 


25 





6 


110 


442 


829 


534 


26 





5 


100 


400 


653 


454 


27 








45 


304 


541 


380 


28 





5 


51 


348 


665 


447 


29 





11 


93 


296 


545 


326 


30 





■1 


37 


253 


535 


388 


31 





6 


39 


296 


597 


419 


32 





1 


32 


210 


460 


363 


S3 


2 


18 


76 


227 


452 


349 


34 







40 


179 


373 


323 


35 







5 


104 


271 


264 


36 




3 


6 


141 


377 


307 


37 




1 


15 


139 


406 


319 


38 







2 


43 


212 


182 


39 







1 


39 


186 


176 


No. Tested . . . 


456 


604 


689 


665 


943 


565 



6o Measurements of Some Achievements in Arithmetic 

TABLE XXVIII 
Location Above Zero of Each Multiplication Problem 



PROBLEM GRADE 


GRADE 


GRADE 


GRADE 


GRADE 


GRADE 


FINAL 


NO. 


III 


IV 


V 


VI 


VII 


VIII 


VALUE 


1 


.256 


.825 


2.017 


2.823 


3.645 


3.886 


1.050 


2 


.245 


.982 


2.094 


3.120 


3.468 


4.181 


1.107 


3 


.633 


1.421 


1.898 


3.016 


3.645 


3.224 


.872 


4 


1.154 


1.626 


2.396 


3.389 


3.645 


3.886 


1.582 


5 


2.187 


2.399 


2.730 


3.166 


3.717 


4.374 


2.380 


6 


2.364 


2.574 


2.730 


3.677 


3.563 


4.004 


2.713 


7 


2.347 


2.268 


2.830 


3.585 


3.942 


4.004 


2.675 


8 


2.394 


2.385 


2.605 


3.534 


3.838 


4.004 


2.616 


9 


2.855 


3.047 


3.276 


4.363 


4.820 


5.078 


3.783 


10 


2.948 


2.998 


3.366 


4.332 


4.805 


5.078 


3.789 


11 


3.725 


3.451 


3.566 


4.504 


4.891 


5.035 


4.089 


12 


3.598 


3.518 


3.711 


4.678 


5.205 


5.602 


4.261 


13 


4.188 


3.999 


4.372 


5.265 


5.618 


5.836 


4.706 


14 


4.891 


4.177 


4.486 


5.562 


5.964 


5.957 


5.046 


15 


4.986 


4.353 


4.456 


4.871 


5.066 


5.138 


4.723 


16 


4.737 


4.041 


4.120 


5.022 


5.596 


6.154 


4.727 


17 




6.297 


5.623 


5.907 


5.429 


5.262 


5.721 


18 


5.234 


4.777 


4.630 


5.438 


5.948 


6.029 


5.242 


19 


5.529 


4.934 


4.672 


5.223 


5.751 


5.915 


5.194 


20 


5.234 


5.134 


5.491 


6.481 


6.799 


6.995 


6.296 


21 




6.558 


5.975 


6.094 


5.618 


5.579 


5.889 


22 




6.910 


6.338 


5.977 


5.611 


5.227 


5.826 


23 


5.097 


5.301 


5.506 


5.346 


5.516 


5.078 


5.375 


24 




5.644 


5.826 


5.619 


5.516 


5.326 


5.625 


25 




6.789 


6.450 


5.834 


5.553 


5.454 


5.825 


26 




6.910 


6.544 


6.083 


6.540 


6.555 


6.290 


27 






7.220 


6.626 


7.011 


7.159 


6.973 


28 




6.910 


7.120 


6.381 


6.489 


6.623 


6.580 


29 




6.450 


6.611 


6.671 


6.996 


7.536 


7.002 


30 




7.614 


7.359 


6.915 


7.038 


7.101 


7.066 


31 




6.789 


7.319 


6.671 


6.784 


6.861 


6.850 


32 




7.614 


7.458 


7.176 


7.332 


7.281 


7.290 


33 


5.529 


6.128 


6.786 


7.074 


7.366 


7.379 


7.069 


34 






7.306 


7.379 


7.679 


7.555 


7.504 


35 






8.618 


7.965 


8.122 


7.947 


8.020 


36 




7.159 


8.481 


7.652 


7.668 


7.660 


7.656 


37 




7.614 


7.961 


7.667 


7.546 


7.581 


7.647 


38 






9.250 


8.711 


8.408 


8.509 


8.533 


39 






9.575 


8.784 


8.552 


8.551 


8.609 



Data From Which Scales Were Developed 6i 

TABLE XXIX 

Distribution According to the Number of Division Problems Solved 

GRADE grade GRADE GRADE GRADE GRADE 

III IV V VI VII VIII 

36 2 6 5 

35 9 21 

34 7 25 33 

33 10 38 47 

32 11 48 59 

31 1 18 74 51 

30 1 29 67 62 

29 25 86 63 

28 3 42 74 41 

27 3 46 77 51 

26 2 40 73 30 

25 7 45 58 29 

24 13 51 58 11 

23 1 20 41 60 11 

22 1 27 42 38 10 

21 1 41 47 28 6 

20 3 49 39 29 7 

19 2 53 40 26 1 

18 4 47 30 8 1 

17 4 50 26 19 

16 14 48 18 14 1 

15 1 10 43 16 8 2 

14 5 19 77 13 6 

13 14 48 48 9 3 

12 7 47 46 11 2 

11 4 74 33 4 1 

10 12 66 21 2 2 

9 12 67 20 1 1 

8 13 58 11 1 

7 15 44 3 2 

6 21 37 2 

5 27 25 4 1 

4 12 25 4 2 

3 12 17 1 

2 10 12 3 1 

1 21 8 3 

32 18 1 

No. Tested... 218 605 685 670 940 542 

Median 5.815 9.873 16.469 23.781 27.442 30.113 

25 Per Cent.. 2.125 7.211 13.401 19.813 23.800 27.519 

75 Per Cent.. 9.042 12.585 19.919 27.489 30.478 32.500 

QUARTILE 3.459 2.687 3.259 3.838 3.339 2.491 



62 Measurements of Some Achievements in Arithmetic 

TABLE XXX 

Number in Each Grade that Solved Each Problem in Division 
Correctly 



problem 


grade 


GRADE 


GRADE 


GRADE 


GRADE 


GRADE 


no. 


III 


IV 


V 


VI 


VII 


VIII 


1 


91 


302 


506 


547 


822 


470 


2 


110 


325 


544 


623 


912 


533 


3 


98 


369 


548 


578 


865 


526 


4 


100 


365 


571 


656 


929 


541 


5 


87 


451 


657 


662 


934 


542 


6 


138 


500 


656 


661 


932 


541 


7 


124 


485 


656 


644 


913 


532 


8 


110 


428 


649 


653 


930 


538 


9 


76 


415 


649 


655 


931 


535 


10 


24 


44 


315 


493 


859 


527 


11 


42 


271 


473 


612 


917 


536 


12 


96 


446 


607 


582 


824 


485 


13 


93 


453 


595 


605 


856 


511 


14 


12 


144 


462 


618 


893 


534 


15 


4 


42 


254 


496 


791 


506 


16 





39 


415 


512 


818 


502 


17 


4 


31 


180 


458 


779 


491 


18 


4 


178 


463 


578 


857 


521 


19 





78 


313 


379 


670 


433 


20 





37 


283 


402 


621 


401 


21 





16 


229 


433 


731 


476 


22 





40 


267 


450 


725 


485 


23 


5 


25 


132 


433 


675 


451 


24 







91 


260 


640 


438 


25 




8 


63 


373 


638 


420 


26 




1 


143 


352 


658 


426 


27 




6 


155 


401 


678 


464 


28 






18 


208 


364 


323 


29 






18 


221 


442 


320 


30 






6 


209 


491 


397 


31 






18 


207 


420 


313 


32 






10 


179 


379 


299 


33 









74 


265 


313 


34 






5 


115 


321 


254 


35 






1 


30 


158 


136 


36 









28 


123 


139 


No. Tested . . . 


218 


605 


685 


670 


940 


542 



Data From Which Scales Were Developed 63 

TABLE XXXI 
Location Above Zero of Each Division Problem 



•ROBLEJ 


I GRADE 


GRADE 


GRADE 


GRADE 


GRADE 


GRADE 


FINAL 


NO. 


III 


IV 


V 


VI 


VII 


VIII 


VALUE 


1 


2.230 


2.937 


3.538 


4.515 


4.823 


5.532 


3.586 


2 


1.900 


1.795 


3.270 


3.550 


3.733 


4.035 


2.563 


3 


2.106 


2.519 


3.244 


4.233 


4.439 


4.392 


3.194 


4 


2.072 


2.546 


3.049 


2.840 


3.176 


2.906 


2.457 


5 


2.299 


1.956 


1.908 


2.509 


2.797 


2.581 


2.083 


6 


1.415 


1.542 


1.925 


2.555 


3.016 


2.906 


1.574 


7 


1.661 


1.674 


1.925 


3.241 


3.711 


4.104 


2.312 


8 


1.900 


2.126 


2.090 


2.950 


3.072 


3.538 


2.182 


9 


2.494 


2.214 


2.090 


2.869 


3.016 


3.881 


2.395 


10 


3.730 


5.088 


4.636 


4.919 


4.496 


4.347 


4.596 


11 


3.205 


3.127 


3.752 


3.839 


3.590 


3.786 


3.484 


12 


2.143 


1.993 


2.699 


4.192 


4.802 


5.322 


3.160 


13 


2.192 


1.937 


2.824 


3.920 


4.525 


4.837 


3.045 


14 


4.289 


3.990 


3.818 


3.752 


4.083 


3.962 


3.958 


15 


5.030 


5.132 


4.976 


4.901 


5.041 


4.947 


4.982 


16 




5.178 


4.088 


4.788 


4.852 


5.036 


4.671 


17 


5.030 


5.358 


5.427 


5.145 


5.113 


5.228 


5.263 


18 


5.030 


3.736 


3.810 


4.233 


4.515 


4.567 


4.058 


19 




4.610 


4.647 


5.609 


5.688 


5.938 


5.304 


20 




5.226 


4.813 


5.483 


5.906 


6.227 


5.564 


21 




5.814 


5.123 


5.300 


5.387 


5.453 


5.357 


22 




5.167 


4.901 


5.195 


5.421 


5.322 


5.157 


23 


4.877 


5.512 


5.773 


5.300 


5.667 


5.754 


5.481 


24 






6.136 


6.277 


5.824 


5.890 


6.038 


25 




6.233 


6.458 


5.642 


5.833 


6.061 


5.911 


26 




7.533 


5.688 


5.762 


5.740 


6.005 


5.782 


27 




6.383 


5.602 


5.483 


5.653 


5.605 


5.579 


28 






7.368 


6.586 


6.948 


6.821 


6.868 


29 






7.368 


6.507 


6.634 


6.844 


6.762 


30 






7.993 


6.582 


6.440 


6.263 


6.428 


31 






7.368 


6.595 


6.720 


6.893 


6.826 


32 






7.745 


6.777 


6.882 


6.987 


6.882 


33 








7.666 


7.377 


6.893 


7.241 


34 






8.130 


7.258 


7.130 


7.296 


7.222 


35 








8.369 


7.949 


8.177 


8.168 


36 








8.417 


8.185 


8.153 


8.227 



Deacidified using the Bookkeeper process. 
Neutralizing agent: Magnesium Oxide 
Treatment Date: Nov. 2004 

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